The absolute safest way to store Moon (or any cryptocurrency) is by using a paper wallet. A paper wallet simply cannot be hacked by any malicious computer offensive. But, as always, if someone gains access to your private key, they gain access to your account. It is also highly recommended to store a paper wallet in multiple locations, in case of fire, theft, etc.
Due to the inherent inconvenience for the average user in storing Moon on a paper wallet, the most popular method of storage is a desktop wallet, followed by a mobile wallet.
Mooncoin desktop wallets are available in both Windows and Mac formats.
Mooncoin mobile device wallets are available here: