A team has finished a game with 13-13 singles and extra-base hits 2 times since 1897, most recently California Angels on 20 June 1980. That is about one every 221,502 team-games.
Box score for the most recent one ↗
13-7 13-8 13-9 13-10 13-11 13-12 13-13 13-14 13-15 14-0 14-1 14-2 14-3
The whole distribution — Singles and Extra-Base Hits · All 18 shapes