In my case I want to tune my guitar, but the app I'm looking for doesn't (or even shouldn't) have to be only for guitars.
Following are my top requirements:
Quick start: therefore it should be lightweight, ideally I just want to open my app and then immediatly play the first string on my guitar to tune it, no other functions like metronome and such needed
Accuracy: it should not wobble around too much (unless of course the sound really wobbles in the human ear), I've had this experience with a couple of apps, so there needs to be a good filter
Automatic tune detection switch: I should be able to either set a certain note I'm trying to reach or just let it recognize my note automatically without a goal
Saveable sets of notes, e.g. 6 different notes of a guitar like many apps have but as soon as you want to have a different tuning or even create your own you have to go pro and pay -.-
Don't tell me similar apps because I've tried out many similar apps already, only if you happen to actually know one that meets at least 4 out of my 5 requirements it might be the one I was looking for :)
Edit: If you think you know a better app, than in the answer feel free to post it. If I think so too, I will mark that as the answer. Otherwise it might still be usable for others who don't have the same requirements as me so I think it would be nice to have a selection of the best tuning apps in the answers as there are many bad ones out there in my opinion.