The corpse is cremated, and the urn with the ash is usually kept in a chedi in the local temple.
A third option, which is seemingly becoming more popular these days, is called “loi angkarn” which means the floating or scattering of ashes over the water. However, they might keep some relics, like pieces of bone, in the shrine at home. It is not really a Buddhist tradition as it has been adapted from Hinduism where they often scatter ashes in the Ganges River. Some Thai people believe that floating the ashes of their loved ones in a river or in the open sea will help wash away their sins but also help them go more smoothly up to heaven. It doesn’t matter where you do this, but if you are in the Bangkok and Samut Prakan area then an auspicious place is the mouth of the Chao Phraya River at Paknam where I live.