This app was mentioned in 2 comments, with an average of 4.50 upvotes
Not sure if you're on iOS or Android but an app for Android I've always used is Tennis Temple. It's not the best looking app and there are ads which you can pay to remove but it gets the job done. Live scoring, order of play, Official and live ranking. Can sort ranks by flags also
I sing the praises of Tennis Temple whenever I can, because it's so well designed and feature packed, from live scores with embedded streams, to rankings, to tournament schedules... Just look at how it displays past matches between two players for instance, I've never seen a more readable, prettier h2h graphic anywhere else. It also sends notifications when a player you follow (by clicking the little bells next to their names) starts a match, and another one with the final score when it ends... etc. It's really great.