You will find in Phuket Town but not in Patong or the tourist beach areas.
There 'used to be' places frequented by lots of Thai people with Thai bands and singers in the 90's but they are long gone.
Not sure though that you would want to go to Phuket Town and head back late.
Places in Bangkok, Pattaya, Hua Hin, Hat Yai etc all have places frequented by lots of Thais and have the Thai rock and pop bands, singers, dancers, whole atmosphere but apart from Phuket Town itself that is all you will have.