It's the image everyone knows best – so well known that we're not even going to use it as an illustration here: a Jack Russell, stuffed, standing up, holding in its paws a sign saying "I'm dead". There is also a version in which the stuffed animal is a cat.

Several things strike one when looking at this. The first reaction – I've tried this out on a few people who have somehow managed not to see the image yet – is laughter: a short, shocked laugh that suddenly evaporates, like a drop of water on a hot shovel, as the work's various contradictions and ambiguities align and realign themselves within your consciousness.

First, you notice the audacity. It's a work of what seems like blinding obviousness. But in attributing the ability to express a condition to something that is manifestly unable to do so, Shrigley is having a go at the infantilising anthropomorphism currently sloshing around daily culture: the coffee cup which has "Careful – I'm hot!" printed on it; or, as I saw recently on a tourist double-decker the other day, "Sorry – I'm not in service". But there's more.

What the work is inviting us to do is, literally, to laugh at death – for that is what you are seeing: almost all you are seeing. But not all, for a living hand arranged the body, wrote on the sign, and stuck the sign in the paws. There is life there, but a cruel kind of life, the kind that is rumoured to make sport of the corpses in the back rooms of undertakers, that (at its most innocent) makes the bodies of the dead assume unnatural positions, or look as though they're doing silly things. But there's still more, yet another flip side: he may be making the animals do things they were incapable of when alive, but they're doing things that cartoon animals have no problem doing and, moreover, the truth they are proclaiming can't be gainsaid. That animal is dead, after all, just as its placard proclaims.

The more one thinks about it the more eloquent a statement about death it seems. For all its ambiguities, sparked off from the simplest of elements and generating a surprisingly rich and accessible range of interpretations, there is, just as one may say about death, no let-out, in the end. It makes Damien Hirst's works of taxidermy, with their endless titles, almost look as though, in comparison, they are evading the issue. (There's a Shrigley cartoon in which a father and son are looking at one of Hirst's flyblown heads in a perspex box. "It's bloody brilliant, son, that's what it is," says a speech balloon, and – as Shrigley has used the same joke in a short film urging us, and governments, to support the arts – you suspect that he really does think it's brilliant.)

Yet one of the most curious things about Shrigley's works of taxidermy is that somehow – and I have not got close up enough to one to see if any trickery has been used, but I would guess not – these animals' faces look, uncannily, as though they have been drawn by David Shrigley. The expression, the unsettlingly expressive blankness characteristic of his cartoon figures' pupil-less eyes, is Shrigleyan. They have become subsumed into his world. Now, that really is clever.

In one of his introductions to Shrigley's collections, Will Self wrote that, once you've looked at enough of his drawings – he gives a figure of a hundred – "there is no plane of reality other than that described by Shrigley." It's a good point, and a testament to Shrigley's genius, which is not a word I use lightly.

On first encountering a Shrigley drawing, one is of course immediately aware that we are in a realm of artistic fluidity. You might even experience, before it's sunk in, a spurt of outrage that anything like this can earn any claim to our attention. And then you might ask: are we looking at a cartoon, or a work of art? Surely something so rudimentary cannot be art? But then you can't really say they're cartoons either, or not with complete confidence. A cartoon that does not make us laugh can be said to have failed; a Shrigley drawing that does not make us laugh makes us do something else – think, probably. This already puts it in the premier league of conceptual art, which, too much of the time, makes us only think darkly about Arts Council funding, or the limits of human gullibility.

And yet by adopting the aesthetic of the disturbed adolescent who can't draw particularly well, or the disturbed man in a pub toilet with a pen, a blank surface to draw on and a bit of time on his hands, Shrigley sneaks profundity in under the radar. He is adept at blurring boundaries, as everyone who thinks about him notices: "naive/sophisticated; whole/part; framed/unconstrained; to scale/in perspective; naturalism/fantasy" (Self again). To which one can add, among other things: funny/not funny.

He also, in his sculpture, make us wonder whether we are in fact seeing a sculpture or a three-dimensional cartoon. One of my favourites is a cardboard box, placed on some cleared and derelict urban space, perhaps an old bombsite, in what looks like Glasgow. There is a rectangular door-shaped hole cut in the box – which itself looks as though it is roughly four feet across and two feet high, maybe less. Above it are written the words "LEISURE CENTRE". Now, everyone who sees this laughs; and the more you think about it, the richer that laughter is. For something that looks as though it took half a second to dream up, and maybe 60 seconds to execute, this is quite an achievement.

But the slapdash nature of Shrigley's work is deceptive. Winningly happy to talk about his work and his creative methods, he is emphatic about not making too many bold claims. About his libretto for an opera, staged last year, Pass the Spoon, a bizarre story involving a spoon, a fork, a banana, a manic-depressive egg and the sinister Mr Granules (why, incidentally, is that such a great name?), he said: "I suppose that these characters and these events that I've imagined will come from the same place as all the other crap I've produced ... To be honest with you, the only thing I'm really qualified to do is to make the poster." And yet he puts the work in – he spends eight hours a day drawing.

Here are the words I removed to create the ellipsis in the quote above: "they will be recognisable, and I think you will see my hand in it". Sandwiched between two very self-deprecating statements – can you imagine any other artist saying anything like that? – is the acknowledgement of an artist who knows what he's doing, and what he's about. Even when he's not producing art/cartoons, or cartoon/sculptures, he can do something Shrigleyesque. Looking at his work makes us wonder about style, or what it is about an artist's vision that makes it recognisable; how you can see the artist's hand in it.

For an entertaining half-hour, you could do worse than type the words "David Shrigley" into Google and then click on "images". You will get – for Shrigley would appear to be generous with his talent, and would probably knock something out for you if you asked nicely enough – at least 11 pages of cartoons (or whatever they are). "SORRY I PAINTED THE WORD TWAT ON YOUR GARAGE DOOR" is the entire text of one of his image-less drawings (or whatever you want to call them); "PLEASE EXCUSE THE TERRIBLE INJUSTICE" (and in much smaller capitals, below: "THANK-YOU") is that of another. What is it that makes us accept that one sensibility alone produced both of these? What is the place in Shrigley's head to which he alludes that produces this "crap"?

"Our favourite exponent of contemporary outsider art", was how Esquire magazine described him last year, but it is not exactly outsider art (which tends to involve some kind of pity, or condescension, on the part of the viewer). This is, in fact, almost completely wrong: the thing about Shrigley is that he produces insider art: manifestations and expressions of an interior weirdness to which he grants us access, and which we can, at some inarticulate but immediate level, identify with and understand. In the vile and unending struggle against futility, shame and violence, you gather pretty quickly that Shrigley is on your side. It is not an idle exercise. One of the images that will come up in your Google search is what I gather is a tea-towel with these words on it: "TELL ME WHEN I AM NO LONGER NEEDED AND I SHALL GO". To which one can only reply: you're still needed. Do please stick around.