Before Licensed to Ill was released in November 1986, they opened up for Madonna in 1985, they played Orlando, Tampa, and the good ol' Hollywood Sportatorium. They also played Tampa and Miami (and later West Palm) in mid-1986 on the Run-DMC "Raising ****" tour. The West Palm show was right as I was starting my freshman year, some of us wanted to go to see Run-DMC but we had Orientation and move-in and whatnot, plus we were making runs to Home Depot to buy stuff to build lofts in the dorm rooms.
On the Licensed to Ill tour, just a couple months after the album was released, they went to St. Pete, Jax, and the good ol' Sunrise Musical Theater during February. In the summertime, they went to Tally, Tampa, Orlando, Jax, and good ol' Bobby Maduro Stadium in Miami.
Then, if you remember, they had a long lay-off until they put out Paul's Boutique in 1989, and the album notoriously bombed when it was first released. The Beasties played at DECOS on Miami Beach (now THAT club is a blast from the past). I think it was one of only a handful of dates that they EVER did for Paul's Boutique. I know that they HATED the debauchery of the Licensed to Ill tour, and they really didn't tour again until Check Your Head in 1992.
They played club dates in May 1992 as well for Check Your Head. I saw them at The Edge in Orlando, and they played the Cameo in Miami Beach (plus St. Pete). In November 1992, they played The Edge again, Bayfront in Miami, plus Tampa and Jax.
They did Lollapalooza in 1994 when Ill Communication came out. After that, they played mostly arenas, stadiums, and festivals for the rest of their career.