In order to be able to be used in Underdog Softball the bat needs to have either an ASA Certification or a USA certification (not to be confused with a USSSA certification) in order for it to be legal. 

ASA softball has re-branded themselves as USA softball. 
USSSA is another organization. Bats with a USSSA certification, but no ASA/USA certification are not allowed. 

Additionally, if it has an ASA certification stamp, it needs to not appear on their banned bats list, which can be found here:

https://usa.asasoftball.com/e/BB1P2000.asp