A team has finished a game with 4-13 singles and extra-base hits 2 times since 1897, most recently St. Louis Cardinals on 26 June 2016. That is about one every 221,502 team-games.
Box score for the most recent one ↗
3-13 4-8 4-9 4-10 4-11 4-12 4-13 5-7 5-8 5-9 5-10 5-11 5-12
The whole distribution — Singles and Extra-Base Hits · All 18 shapes