Example of non-custodial and open source mobile wallets:
- Electrum
- MyCelium
- Bread Wallet
- Trust Wallet (IOS)
Neat little guide.
To all users not too much tech savy, i would advice you to always choose open source wallets. They are more secure and more verified (so trusted) than closed source projects.
In Bitcoin security is almost everything in terms of using it, so no wallet should be closed source in reality.