Firstly, exchange is not a wallet, many people keep their fund on exchange for one reasons or another. Funds kept on exchanges eg centralized exchanges is exposed to high risks than keeping them on external cryptocurrecy wallets.
In my opinion, these wallets is best to safe keep your coins/tokens and they have no records of any security challenges. You considered using the the following wallets,
Metamask, Trustwallet, Coinbasewallet etc. There are other forms of wallet that can be used to store cryptocurrecy offline like cold wallets, but it all depend on your holding strategy.