A team has finished a game with 22-6 singles and extra-base hits 2 times since 1897, most recently New York Yankees on 12 August 1953. That is about one every 221,502 team-games.
Box score for the most recent one ↗
21-9 21-10 22-0 22-1 22-3 22-4 22-6 23-0 23-2 23-4 23-5 23-6 23-10
The whole distribution — Singles and Extra-Base Hits · All 18 shapes