Royal Festival Hall

This hulking presence on the South Bank was built for the 1951 Festival of Britain. Sixty years on, it's still at the forefront of London's cultural scene. Although primarily a hall for major concerts, it's a pretty safe bet that the ground-floor stage area will be busy with a free concert, exhibition or workshop, especially if you visit on a weekend. While you're there, be sure to take the singing glass elevator up to the fifth floor, where you'll find a little-known balcony area with impressive views of the Thames. There's free Wi-Fi throughout the building, too.

• Southbank Centre, Belvedere Road, SE1,020-7960 4200, southbankcentre.co.uk

Somerset House

The famous courtyard and vast Georgian building alone are worthy of a visit, but seek out the two elegant staircases, the balcony views of the Thames, and the small basement gallery. In the summer, the dancing courtyard fountains are a joy. In the winter, an ice rink takes over. Occasionally, the house's catacombs are opened for art installations and performance. The house also contains the Cortauld Gallery of impressionist paintings, though an entrance fee applies here.

• The Strand, WC2, 020-7845 4600, somersethouse.org.uk. Open Mon-Sun 10am-6pm

BFI Mediatheque

The British Film Institute's swanky South Bank complex has much to explore, including two excellent bars, cinema screens, a well-stocked film store and a small exhibition space. Its greatest treasure, however, is the Mediatheque. This suite of wide-screen computer booths offers free access to thousands of archive TV shows, films and documentaries, including plenty of material about London itself. Book ahead or simply turn up at a quiet moment.

• Belvedere Road, London, 020 7928 3535, SE1, whatson.bfi.org.uk/Online/default.asp. Open Tue-Fri 12 noon-8pm, Sat-Sun 12.30pm-8pm

Wallace Collection

Just north of the exhausting crowds of Oxford Street stands Hertford House, home to the Wallace Collection. The sizeable galleries are noted for the fine paintings, displays of weapons and armour, and elegantly furnished rooms. Painters represented include Rembrandt, Titian and Van Dyck, and the Wallace is also home to the famous Laughing Cavalier by Frans Hals. The restaurant in the central atrium must rank among London's most exquisite gallery dining spaces.

• Hertford House, Manchester Square, W1, 020-7563 9500, wallacecollection.org. Open Mon-Sun 10am-5pm

Skateboard park

Photograph: Linda Nylind/Guardian

The South Bank isn't entirely given over to professional concerts and high-brow culture. A popular skate park lurks beneath the Queen Elizabeth Hall, with enough ramps and raised surfaces to keep a couple of dozen skateboarders and BMX riders happy. It's also a popular spot for graffiti artists who can hone their skills without being collared. Even if you're not a skater, the spectacle is worth a diversion. Nearby, on the north-east pontoon of Hungerford foot bridge, is a "skateboard graveyard", into which broken decks are pitched.

• Southbank Centre, Belvedere Road, SE1, 020-7960 4200, southbankcentre.co.uk

The Scoop

Although many of London's museums and galleries are free, it's a little harder to find theatre or cinema that doesn't come with a ticket price. One hotspot for gratis performance is The Scoop. This sunken space seats 800 and can be found beside City Hall, close to Tower Bridge. During summer months, the amphitheatre is in use almost every evening, hosting live music, plays, film screenings and keep-fit classes. While you're in the area, take a look inside City Hall where a spiral ramp leads down to small gallery spaces, a cafe and a giant map of London. The security can look a bit fearsome, with airport-style scans – but don't worry, you have every right to go in.

• City Hall, 110 The Queen's Walk, SE1, 020-7403 4866, morelondon.com/scoop.html

Sir John Soane's Museum

Photograph: Alamy

The architect of the Bank of England had a magpie's eye for unusual and exquisite bric-a-brac. His former home has long served as a museum space to show off the collection, which includes period furniture, paintings by the likes of Hogarth and, most memorably, the sarcophagus of Seti I. Although it's often cited as a secret small museum, it's actually quite well known, and queues can form for the popular candlelit evenings, held on the first Tuesday of each month, from 6pm-9pm. You'll never see a museum quite like this anywhere else, though.

• 13 Lincoln's Inn Fields, WC2, 020-7405 2107, soane.org. Open Tue-Sat 10am-5pm

Hunterian Museum

Just across the square from Sir John Soane's Museum lurks another free cultural centre. The Hunterian Museum is named after John Hunter, one of the first people to apply scientific method to surgery. Hunter's collection of skeletal remains, diseased organs and other anatomical curiosities is not for the squeamish, but those with a fascination for the human body will find much to get their teeth into, so to speak.

• Royal College of Surgeons, 35-43 Lincoln's Inn Fields, WC2 020-7869 6560, rcseng.ac.uk/museums. Open Tue-Sat 10am-5pm

Postman's Park

Photograph: Alamy

This tiny park within the Square Mile is something of a cliche in lists of "secret" or "unusual" things to do in London, but its recommendation bears repeating. The reason for its popularity lies in a Victorian memorial to people who died while trying to save others. This wall of tragic heroes was created in 1900 by George Frederic Watts. The memorial was recently updated for the first time in 78 years, with the addition of a plaque for Leigh Pitt, who died rescuing a drowning child in 2007.

• King Edward Street, en.wikipedia.org/wiki/Postman's_Park

Grant Museum of Zoology

The Grant Museum must rank as first among equals in London's generous offering of quirky museums. It serves as a teaching museum for zoology students at University College London, but is also open to the public. The recently updated premises are crammed with zoological curiosities, including a bisected pregnant cat, a giant penis bone from a walrus and a jar full of pickled moles. The museum is also noted for its regular free screenings of cult or forgotten movies featuring animals or monsters, usually followed by a complimentary glass of wine and tour of the museum.

• 21 University Street, WC1, 020-3108 2052, ucl.ac.uk/museums/zoology. Open Mon-Fri 1pm-5pm

Ceremony of the Keys

Every night, for something like 700 years, the Yeoman Warders of the Tower of London have performed a gate-closing ritual known as the Ceremony of the Keys. Only once, when a bomb knocked a couple of warders off their feet, has the ceremony been so much as delayed. It has never been cancelled. Members of the public can view the ceremony for free by writing to Ceremony of the Keys Office, Tower of London, London, EC3N 4AB.

• Tower of London, EC3, 0844 482 7777, hrp.org.uk/TowerOfLondon. Open Tues-Sat 9am-5.30pm, Sun-Mon 10am-5.30pm

British Museum

Photograph: Alamy

It would be churlish to leave the British Museum out of a list like this, even though its fame no doubt precedes it. Most visitors will make a beeline for the controversial Elgin Marbles – statues removed from the Parthenon in Athens – by way of the ever-popular Egyptian displays. But to say there's much more to the museum would be an understatement. The endless series of galleries contain artefacts from just about every major civilisation on earth. The Enlightenment rooms in the east wing are particularly fascinating, and usually crowd-free. And, oh, that magnificent glass roof.

• Great Russell Street, 020-7323 8299, WC1, britishmuseum.org. Open Sat-Thur 10am-5.30pm, Fri 10am-8.30pm

Royal Institution

It's a pretty amazing building that can claim 14 Nobel Prize wins and the discovery of ten chemical elements. The Royal Institution has, for more than 200 years, served as a leading centre of science, nurturing the careers of Michael Faraday and Humphry Davy. Although its regular lectures normally involve a fee, the small basement museum is always freely open. Here you can see various contraptions and instruments from the RI's past – including the world's first thermos flask – and watch genuine scientists going about their business in the nanotech lab.

• 21 Albemarle Street, W1, 020-7409 2992, rigb.org. Open Mon-Fri 9am-6pm

All Hallows By The Tower

While millions of tourists flock to the Tower of London, another historic building lurks just yards away. The church of All Hallows dates back to Saxon times. Although the current building is largely a post-war reconstruction, you can still see remains, including an arch, from more than 1,000 years ago. The crypt contains a small museum, including a model of Roman London and the baptism record of William Penn, founder of Pennsylvania.

• Byward Street, EC3, 020-7481 2928, allhallowsbythetower.org.uk. Open Mon-Fri 8am-6pm, Sat-Sun 10am-5pm (except during services)

Barbican Centre

Photograph: Graeme Robertson

A strange tip, this one, but go and get yourself lost in the Barbican estate, a 1970s complex of baffling walkways, hidden gardens, slopes and steps. The estate is one of Britain's best expressions of the architectural style known as brutalism. Traffic is banned and the pedestrian is king. It's also a complete maze (hence the ironic sculpture of a minotaur). Eventually, though, you'll find the picturesque pond area in front of the Barbican Centre. This arts venue often puts on free entertainment. Its Curve Gallery is always free to visit and specialises in oddball but approachable exhibitions. A rooftop conservatory is another highlight.

• Silk Street, 020-7638 4141, barbican.org.uk

Guildhall Art Gallery

Although most of London's galleries are either in the West End or East End, one of its most rewarding can be found sandwiched between the two, within the Square Mile. Guildhall Art Gallery contains an impressive collection of canvases and sculpture, many depicting scenes from the City's history. A statue of Margaret Thatcher, made of stone not iron, stands guard in the corner. As an added bonus, the basement houses the remains of the Roman amphitheatre, discovered beneath the gallery in 1988.

• 5 Aldermanbury, EC2, 020-7332 3700, cityoflondon.gov.uk. Open Mon-Sat 10am-5pm, Sun 12pm-4pm

Tate Modern

Photograph: Dan Chung

Little more than a decade old, Tate Modern is now firmly established as one of the world's great modern art galleries. It's housed in the shell of the Bankside Power Station, designed by Sir Giles Gilbert Scott, who also gave us the red phone box and Battersea Power Station. Inside, the cavernous Turbine Hall is used to good effect. Every six months, a leading artist fills the space with a giant sculpture, such as a monumental glowing sun from Olafur Eliasson and the 100-million hand-painted "seeds" from Ai Weiwei. The rest of the gallery shows contemporary art from 1900 onwards. A major new wing is currently under construction.

• Bankside, SE1, 020-7887 8888, tate.org.uk/visit/tate-modern. Open Sun-Thur 10am-6pm, Fri-Sat 10am-10pm

Gresham College

The college has been doling out free lectures for more than 400 years, since Thomas Gresham set the ball rolling in his will of 1597. The college now arranges 140 talks a year on every topic from law to divinity to astronomy. Although the organisation is styled as a college, and lectures are normally delivered by professors, it's a very open place aimed at the general public. Most of the events come from within the charming Barnard's Inn Hall, parts of which date back to before even Gresham's time.

• Barnard's Inn Hall, EC1, 020-7831 0575, gresham.ac.uk/lectures-and-events

St Bride's Church

A death knell for a nave? … the steeple of Sir Christopher Wren's St Bride's church

Nicknamed "The Journalists' Church", thanks to its associations with nearby Fleet Street, St Bride's has a much older history than the printed word. A place of worship has probably stood on the site since the 7th century AD. The current church was built in 1672 by Christopher Wren following the great fire of London. Its famous steeple is said to have inspired the first tiered wedding cake, in happy resonance with the church's name. The crypt conceals a fascinating exhibition of the church's past, including Roman remains and medieval glass.

• Fleet Street, EC4, 020-7427 0133, stbrides.com, Mon-Fri 8am-6pm, Sun 10am-6.30pm

National Portrait Gallery

Photograph: Linda Nylind/Courtesy of National Portrait Gallery, London

Nestling up against the National Gallery, the NPG has a peerless collections of historic and contemporary portraits. The Tudor collection is particularly notable, with vast likenesses of Henry VIII, his family and court. The ground floor is also popular, and carries portraits of dozens of modern celebrities and notables. The cafe on the top floor (sadly, not free) offers impressive views of the West End.

• 2 St Martin's Place, WC2, 020-7306 0055, npg.org.uk. Open Sat-Wed 10am-6pm, Thur-Fri 10am-9pm

New London Architecture

Loom Godzilla-like over a giant model of London at this ever-excellent temple to the capital's built environment. The plastic model stretches a dozen metres, taking in the Docklands and Olympic Park in the east, and stretching as far as Battersea in the west. The walls, meanwhile, are plastered with information about proposed new buildings for the capital. This place often gets left out of the guidebooks, but it's one of the best starting points for getting an overview of London.

• The Building Centre, 26 Store Street,WC1, 020 7636 4044, newlondonarchitecture.org. Open Mon-Fri 9.30am-6pm, Sat 10am-5pm

Speakers' Corner

Photograph: Alamy

Any day of the week, there's free entertainment to be had in the north-east corner of Hyde Park. You can even join in. Speakers' Corner has existed since late Victorian times, and welcomes a motley assembly of orators who tend to cover political or religious subjects. Anyone can turn up and declaim on any subject, so long as they stay within the laws of free speech.

• North-east corner of Hyde Park, near Marble Arch, W1

National Gallery

Broadcaster Clive James once described the National Gallery as "the best free show in town". It still is. Some sections, such as the impressionists, can be uncomfortably busy, but the building is so large that you can always find a quiet room full of treasures. Notable paintings (and it's hard to pick just three) include Van Gogh's Sunflowers, Turner's The Fighting Temeraire and Holbein's The Ambassadors. The gallery opens late on Friday evenings, offering free tours and talks.

• Trafalgar Square, WC2, 020-7747 5958, nationalgallery.org.uk. Open Sat-Thur 10am-6pm, Fri 10am-9pm

Museum of London

Both tourists and dyed-in-the-wool Londoners will find plenty of interest at the Museum of London. Recently refurbished galleries tell the story of the city from its prehistory through to the modern day. See the Mayor's golden coach, visit a Georgian pleasure garden and walk through the ever-popular Victorian shopping arcade. Londonophiles will also have a field day browsing the museum's superb bookshop.

• 150 London Wall, EC2, 020-7001 9844, museumoflondon.org.uk. Open Mon-Sun 10am-6pm

British Library

The orange-brick headquarters of the BL aren't just for researchers and scholars. The titanic complex houses plenty of diversions that anyone can visit for free. First among them is the Treasures exhibition, which displays literature as diverse as Magna Carta and original Beatles lyrics. Two small galleries off the main foyer also present ever-changing temporary exhibitions on literary themes. A quirky bookshop and excellent cafe with free Wi-Fi add to the attractions. And don't miss the sculptures by Antony Gormley and Eduardo Paolozzi in the forecourt.

• 96 Euston Road, NW1, 01937 546546, bl.uk. Open Mon, Wed-Thur 9.30am-6pm, Tue 9.30am-8pm, Fri 9.30am-6pm, Sat-Sun 11am-5pm

The Bank Of England Museum

It may not explain how the banking profession got into the mess it's in now, but the museum dedicated to the profession is still worth a visit. Vintage bank notes, art work and antique furniture from the bank make for a small but dense collection. Current exhibitions include one on Gold – and the opportunity to lift up a solid gold bar (ensconced in a cage for obvious reasons); and a display of claims for the replacement of Bank of England notes from the relatives or legal representatives of victims of the Titanic disaster, from survivors and from members of the public who had sent banknotes in letters carried aboard the liner.

• Bank of England, Threadneedle Street, EC2R, 020 7601 5545, bankofengland.co.uk

Matt Brown is editor of the Londonist

• This article was amended on 24 July 2012 to remove a section about mudlarking on the advice of English Heritage and other bodies which are concerned with the protection of the foreshore and London's archaeological heritage. The Bank of England Museum entry was added on 26 July 2012.