A team has finished a game with 3-12 singles and extra-base hits 4 times since 1897, most recently Toronto Blue Jays on 26 July 2013. That is about one every 110,751 team-games.
Box score for the most recent one ↗
2-11 2-12 2-13 3-9 3-10 3-11 3-12 3-13 4-8 4-9 4-10 4-11 4-12
The whole distribution — Singles and Extra-Base Hits · All 18 shapes