Aegis Authenticator
Android
2FA Code Vault
Aegis Authenticator is a free, secure and open source application for Android to manage your 2-step verification tokens for your online services. As it's open source, it is very trustworthy and secure, as seen with the code.
Notes:
Aegis takes additional security precautions by storing your one-time passwords/codes in a vault. In order to view them, you must enter your vault password or use biometrics. This prevents malicious apps from grabbing your codes.
Aegis is not available for iOS. For authenticators supporting iOS, please see other recommendations listed below.
andOTP
Android
2FA Code Vault
andOTP implements Time-based One-time Passwords (TOTP) like specified in RFC 6238 (HOTP support is currently in beta testing). Simply scan the QR code and login with the generated 6-digit code. andTOP only supports Android.
Notes:
andOTP, like Aegis, is a great option. They're both awesome, open source and are developed for Android (not iOS). If you're looking for iOS authenticators, please keep scrolling.
Tofu Authenticator
iOS
Uses iOS Keychain
Tofu is an easy-to-use, open-source two-factor authentication app designed specifically for iOS. It features QR code scanning, account name search ability, support for both counter-based & time-based 6-8 digit codes, and much more.
Notes:
Finally, an authenticator for iOS. We would like to note that Tofu works offline, and stores account details in the iOS keychain. Tofu is also completely open source, with the code being available on GitHub