Let me write these down in a separate post, so I have something to link above.
IO (App dei servizi pubblici)
Used to work, now complains about the Android system being too old (Xperia 10 II, 4.5.0.19 with MicroG).
Tap&Park
Works for me (Xperia 10 II, 4.5.0.19 with MicroG).
Trenitalia
Occasionally complains about missing Google services but otherwise works fine, including ticket purchases. (Xperia 10 II, 4.5.0.19 with MicroG).