A team has finished a game with 19-4 singles and extra-base hits 14 times since 1897, most recently Chicago White Sox on 9 July 2013. That is about one every 31,643 team-games.
Box score for the most recent one ↗
18-9 18-10 19-0 19-1 19-2 19-3 19-4 19-5 19-6 19-7 19-8 19-9 19-10
The whole distribution — Singles and Extra-Base Hits · All 18 shapes