I am writing this at the time of the 5494985 block. The snapshot should give way to some interesting competition for the ETCDEV team. While a lot of controversy surrounded the DAO hack, this should not be confused with a hard fork. What is interesting about Callisto is that it aims to fix security issues that are present in ETC.
Callisto aims to make its blockchain more secure by implementing self-governance and self-funding, which will include protocol to compensate auditors of its code in order to ensure that peer review is not being ignored by Callisto developers. This is an interesting proposal, and it is surprising to be coming from Callisto's team rather than ETCDEV considering that Charles Hoskinson has somewhat of a pioneer of the 3rd Generation, self-governance model.
Callisto claims that the Classic team is a bit too conservative, not addressing security issues and working on functionality like cold-staking. That being said, the team of CLO developers are still planning on starting a smart contract department that will benefit both communities, CLO and ETC, since both development teams will be able to sign off on newly developed smart contracts.
While a reddit post claims that these wallets are supported, https://www.reddit.com/r/EthereumClassic/comments/7qs9ui/callisto_network_announcement/
Guarda Wallet
Trust Wallet
ClassicEtherWallet
ClassicMask
MyEtherWallet
The official temporary website of the CLO dev team only cites that two wallets will support forked coins officially, so if you want to have the smoothest time claiming, maybe use these: Classic Ether Wallet and ClassicMask.
Of course, theoretically any wallet that allows you to export private keys could be used to claim Callisto, doing so usually means that to be absolutely safe, you empty the wallet you were holding at the time of the snapshot and then burn the wallet after exporting the private keys, which can be somewhat of a hassle.
Also note, I have just received notice from YOBIT that they too will be supporting CLO coins, however, withdrawal of ETC will be halted for 5-7 days while they secure wallets against replay attacks. I would recommend that anyone attempting to claim CLO waits, or at least does a smaller test transaction, before spending any ETC that they are looking to unload after claiming CLO. This might be a good option for anyone who plans on unloading the CLO coins right away, but I think this coin makes a strong case for HODLing.
By the time I am finishing up this quick article, block height has reached 5495074, in roughly 15 min... just saying. Good luck Callisto Development. The community is hoping you have a smooth Airdrop. Thanks.