Authy, from what I understand, requires a phone number as backup, meaning it could be compromised by the same method
Google authenticator can't be backedup, which is royally annoying when you change/lose devices
Lastpass has some security issues, and one well known comment here has recommended no one use it.
I heard someone say they use Duo security: does that let you store codes? When I looked it just seemed to lock various services