If you don't want to read the whole list, and just want to choose a Bitcoin Cash wallet...

(SPV = Simplified Payment Verification, does not depend on a server so it's a more decentralized.)

Sorted by various things

For PC/MAC

Electron Cash (a.k.a Electrum for Bitcoin Cash)

For Android

CoPay (SPV)

BTC.com

UnitWallet (SPV)

For iOS

CoPay (SPV)

Bitcoin.com (SPV) (updated rarely)

BTC.com

*CoPay uses a different address format. Not to be confused, please read this

Major Full Nodes

Bitcoin Unlimited Cash Edition

One of the most used cash full node wallet softwares. It has XThin support that saves bandwith while downloading new blocks and uses Parallel Validation to validate blocks faster and safer.

Website: www.bitcoinunlimited.info

Download Cash Release: www.bitcoinunlimited.info/download

Bitcoin ABC

First Bitcoin Cash full node wallet software. The most popular Bitcoin Cash node as it's the first.

Website: bitcoinabc.org

Download: download.bitcoinabc.org

Bitcoin XT

First Bitcoin scaling software - first version released in 8/2015. It supports XThin, and it supports XThin's alternative Compact Blocks too - making it (theoretically) faster and bandwidth-cheaper while downloading new blocks than other clients. It's the least used Bitcoin Cash full node, for no reason.

Website: bitcoinxt.software

Download: https://github.com/bitcoinxt/bitcoinxt/releases/latest

If you want to run one of the full-nodes but don't want to store the blockchain (140 GB)

You can use pruning, by pruning you can choose how much space allocate to these full node softwares. It supports both Bitcoin and Bitcoin Cash.

Other Major/Semi Major Wallets

Electron Cash (lightweight wallet)

The only major talented lightweight wallet - doesn't store the entire blockchain on your computer. It's a changed version of Electrum, one of the most used Bitcoin wallets.

Website & Download: electroncash.org

WARNING: There are fake Electron Cash wallets that steal your money. It is safe if and only if you're downloading it from www.electroncash.org!

BTC.com Wallet (online wallet - mobile)

The most used online wallet Bitcoin Cash & one of the best mobile Bitcoin Cash wallets

Use it here: bcc-wallet.btc.com

Google Play: https://play.google.com/store/apps/details?id=com.blocktrail.mywallet

App Store: https://itunes.apple.com/us/app/btc-com-bitcoin-wallet/id1019614423

Blockchain.info (online - mobile)

One of the major online wallets (most used?) that completely supports Bitcoin Cash. https://cointelegraph.com/news/blockchaininfo-releases-full-bitcoin-cash-support-users-receive-coins

Note: that for anyone opening a new wallet or for anyone who didn't have BTC before 1st August in their wallet, you cannot use it for storing BCH. https://support.blockchain.com/hc/en-us/articles/115005180866-Can-I-use-bitcoin-cash-BCH-in-a-new-or-existing-wallet-

Coinbase (online - mobile)*

One of the major online wallets. They are going to add option to withdraw all of your Bitcoin Cash on 1 January 2018.

Announcement: https://support.coinbase.com/customer/portal/articles/2853600-bitcoin-cash---frequently-asked-questions

Bread [former name: Breadwallet] (mobile)

Breadwallet is the safest iOS Bitcoin wallet with Bitcoin Cash one-time-withdrawal support. It doesn't use a server, instead it connects to three peers, making it more decentralized. There is a rumour that they are going to add an option to switch to Bitcoin Cash and use it like the normal Bitcoin wallet. [noref]

Announcement: https://breadwallet.com/blog/breadwallet-plan-bitcoin-cash-bch/

CoPay - BitPay (mobile - lightweight - Chrome extension - desktop)

Bitcoin Cash full support has been released, but it uses a new address format.

Announcement: https://blog.bitpay.com/bitcoin-cash-wallet/

Coinomi (mobile)

Coinomi is one of the most known Bitcoin Cash wallets (actually it supports over 70 cryptocurrencies), however it may not be a good idea to have more than 50$, as it went closed source last year. Their developers say on its website that the private keys don't leave your device.

Android: https://play.google.com/store/apps/details?id=com.coinomi.wallet

iOS: Coming in future

Armory

One of the good Bitcoin software. Allows you to withdraw BCC (BCH, Bcash on Armory). You should read instructions below.

If further hard forks happen on BCC, its developers won't update Armory to be compatible with it. (Source)

Instructions: https://bitcointalk.org/index.php?topic=2070058.0

Mycelium

Actually Mycelium can't do anything about your Bitcoin Cash, but you can use BTC.com's tool to claim it

https://blog.btc.com/how-to-recover-your-trapped-bitcoin-cash-from-other-wallets-24b929543f26

Or you can claim it using Coinomi

https://steemit.com/btc/@libert/step-by-step-guide-on-claiming-bitcoincash-from-mycellium-wallet-201783t112141477z

Bitcoin.com wallet (mobile - desktop - Chrome extension)

A good option as a wallet. It's changed version of CoPay that fully supports Bitcoin Cash and runs on Windows, Mac, Linux, iOS, Android.

Download: www.bitcoin.com/choose-your-wallet/bitcoin-com-wallet

Jaxx (mobile, Chrome extension)

Jaxx has full Bitcoin Cash support (Android and Chrome extension).

Announcement: https://www.reddit.com/r/btc/comments/7bu6yo/jaxxio_now_supports_bitcoin_cash/

TSBW (online)

It's an open source wallet that can be used without opening an account. The easiest software to use with paper wallets.

WARNING: Developer accused of stealing funds. More info 1 More info 2

Bitcoin Cash: https://tsbw.io/bcc/

BitcoinCashTribe (online)

Works similar to TSBW.

Website: https://bitcoincashtribe.com/index.php

BitPie & Bither

They have full Bitcoin Cash support.

BitPie announcement: http://docs.bitpie.com/en/latest/bitpiebcc/index.html

Bither announcement: http://docs.bitpie.com/en/latest/bitherbcc/index.html

Exodus (lightweight)

One of the popular wallets that support Bitcoin Cash 100%.

Announcement: https://redd.it/766ojz

Download: https://www.exodus.io/releases/

Xapo (mobile)

You can withdraw all of your BCH at once before December 14th, or convert your BCH to BTC by one-click.

Announcement: https://blog.xapo.com/time-to-convert-your-bitcoin-cash-bch/

BitGo (online - Chrome extension)

BitGo supports Bitcoin Cash.

Announcement: https://blog.bitgo.com/bch-availability-c252727ea1be

StrongCoin (online)

StrongCoin supports Bitcoin Cash.

Announcement: https://strongcoin.com/blog/bitcoin_cash_is_now_supported_in_strongcoin_wallet

Parity Bitcoin (full node + lightweight)

Parity Bitcoin is a high performance wallet that can be run as a full node or a lightweight client. (Warning: Don't be confused with Parity! Parity Bitcoin is the Bitcoin/Bitcoin Cash version of Parity Ethereum client.)

Announcement: https://blog.parity.io/parity-bitcoin-is-now-supporting-segwit-segwit2x-and-bitcoin-cash/

Bitcore (full node)*

The 5.0 version which is coming soon, supports Bitcoin Cash if pointed to a Bitcoin Cash node. Doesn't have a user interface, not the best choice for a user. Useful APIs. (This is not a wallet for users! This is for developers!)

Announcement: https://news.bitcoin.com/now-bitcore-nodes-can-switch-to-any-bitcoin-implementation-even-forks/

Bitcoin Cash wallet (by Chris Pacia)

Changed version of OpenBazaar's SPVWallet for Bitcoin Cash. It has both GUI and command line support.

Downloads (Windows & Linux): https://github.com/cpacia/BitcoinCash-Wallet/releases/latest

Bitcoin Cash wallet (by HashEngineering) (Android)

Changed version of Andreas Schildbach's Bitcoin Wallet for Bitcoin Cash.

Announcement: https://www.reddit.com/r/btc/comments/72xost/beta_for_bitcoin_cash_wallet_for_android_recover/

StashWallet

A new HD Android wallet that fully supports Bitcoin Cash.

Website: https://stashcrypto.com

Freewallet (mobile)

A Bitcoin Cash Android wallet. WARNING: READ THIS

Website: https://freewallet.org

Google Play: https://play.google.com/store/apps/details?id=bcc.org.freewallet.app

UnitWallet (mobile)

One of the Bitcoin-Cash-only wallets, fork of bread[wallet].

Website: https://www.unitwallet.co

Google Play: https://play.google.com/store/apps/details?id=com.unitwallet

Hardware Wallets

bitWallet (iOS)

A new Bitcoin wallet with Bitcoin Cash and a few altcoins support. It's good to keep in mind that it's not open source.

Website: http://www.sollico.com/bitwallet/

Download: https://itunes.apple.com/us/app/bitwallet/id777634714

Trezor

One of the major hardware wallets. Allows you to claim your Bitcoin Cash (They call it Bcash) if it's updated to latest firmware (1.5.1+). Fully supports Bitcoin Cash.

Trezor Guide: https://blog.trezor.io/claim-bcash-bitcoin-cash-bch-bcc-trezor-wallet-f0a810d5864a

KeepKey

One of the major hardware wallets + fully functioning Bitcoin Cash wallet.

https://coingeek.com/keepkey-wallets-now-support-bitcoin-cash/

Ledger

One of the major hardware wallets. Supports Bitcoin Cash if you update your firmware to latest version.

Guide: http://support.ledgerwallet.com/knowledge_base/topics/bitcoin-cash

Digital Bitbox

Its firmware supports Bitcoin Cash, but the desktop client doesn't. You can claim your Bitcoin Cash by sending all of them to your Bitcoin Cash wallet.

Cannot be used as a Bitcoin Cash hardware wallet.

Guide: https://digitalbitbox.com/bitcoincash

Paper Wallet

The safest Bitcoin/Bitcoin Cash hardware wallet. You can claim your Bitcoin Cash by importing to a Bitcoin Cash software.

You can create a paper wallet on cashaddress.org or bitaddress.org or bitcoin.com

All Bitcoin paper wallets are compatible to Bitcoin Cash as they both use the same private keys/addresses.

The wallets that don't support Bitcoin Cash aren't on the list.

List is sorted by various things.

*: Description will be updated.