MBTA in Boston had the mTicket app for mobile ticketing and payments for years. I live in Boston and use the app regularly. Can't comment on other cities because when I visit for a short trip I typically don't bother installing apps.
Amtrak and most airlines use mobile boarding passes too. Interestingly enough, on my recent trip to Europe I used the mobile boarding pass in Logan airport just like everyone else. But in Frankfurt when I showed my phone to the agent they looked at me like I was from another planet, probably thinking "stipid americans"
And while we're on the subject of transportation, about 5 years ago I visited a bunch of european countries, including my home country in Europe, and at that time the only way to call a cab was via dialing the local phone number, cash only of course. Funny because on that trip heading to the airport in the States was matter of acouple taps in the Uber app.
For around a decade, many cities, example Prague, have accepted text message based payments for public transit. Today most public transit systems have their own apps for payment and ticketing. I can't think of a single U.S. city that does this. They're all exact change only or proprietary ticketing systems.
About the most advanced I can recall, Citi had a short lived tap and pay, NFC based, project in the NYC subway 10 years ago. You still got the 10% metrocard discount. It was ultra proprietary though, Citi cards only.
And then Citi and Amex went and ripped NFC out of all my credit cards for this slow EMV chip. Haha yeah, when will Europe catch up. What we did is catch up with their 3 decade old chip idea.