A team has finished a game with 4-12 singles and extra-base hits 5 times since 1897, most recently Boston Red Sox on 7 May 2017. That is about one every 88,600 team-games.
Box score for the most recent one ↗
3-12 3-13 4-8 4-9 4-10 4-11 4-12 4-13 5-7 5-8 5-9 5-10 5-11
The whole distribution — Singles and Extra-Base Hits · All 18 shapes