Bread wallet is pretty alright for a phone wallet if you're using IPhone. If you're using Android, I would say Mycelium is better.
The security risks of phone wallets mostly aren't with the app itself. The portability of the phone will have you bring it everywhere you go. The downside of it is that you are more prone to losing your phone and having your wallet stolen. Of course, you may have a seed backup and a PIN code but I wouldn't think it is enough.
If you're looking for a lightweight wallet, I recommend Electrum.
+1 for Mycellium. Been using the app together with Schildbach's wallet for 3 years now and great developments especially when they first added the functionality of signing messages using the mobile app itself.
Another thing for mobile phone wallets, you might to consider on limiting the sites you browse and downloading apps using your mobile. Apparently, some websites do have cookies which hijack your session into whatever account you're logged in using your browser and can gain sensitive info about you. Downloading apps is self-explanatory; malicious code can steal bitcoins.