List of #1s achieved on profile
Idea stripped right from osu!
On the profile, there would be a list of levels where the player has achieved the spot with the most score on a level. A bit like this:

Maybe this could be expanded to the server returning count scores by a user for a given rank over the API? A client would be able to freely specify a rank to search for; and for the website, we could probably allow a user to see another user's rank 1, 2 and 3 scores, together with previews of the levels they were achieved on, per score type on seperate pages. This might require caching the scores' ranks similarly to level/user/playlist statistics, else the server would have to iterate through count amount of leaderboards for a single request every time to find out which of the user's scores have the given rank.
I personally don't think we/anyone else would really need anything other than 1, but that sounds like a small amount of effort so we can go ahead with that. My only request is have 1 be the default and have the option for it be not-required.
A table for score statistics sounds good for this.