A team has finished a game with 5-14 singles and extra-base hits 1 time since 1897, most recently Colorado Rockies on 31 May 2016. That is about one every 443,004 team-games.
Box score for the most recent one ↗
5-8 5-9 5-10 5-11 5-12 5-13 5-14 5-15 5-17 6-6 6-7 6-8 6-9
The whole distribution — Singles and Extra-Base Hits · All 18 shapes