Familysearch.org has an index for men who served in the War of 1812. The database description is as follows:
"Name index to compiled military service records of volunteer soldiers who served in the War of 1812. The files are located in the National Archives in Washington D.C. This collection is a part of RG 94, Records of the Adjutant General's Office, 1780's-1917 and is National Archive Microfilm Publication M602."
I found two listings in this database in the section "Sim-Siz."On page 2325, there is a Thomas Singleton who served in the 2nd Regiment (Jennings) KY Volunteers.On page 2326, there is a Thomas Singleton who served in the 15th Regiment (Slaughter) of the KY Militia.I believe that the commander of the 15th Regiment (Slaughter) was Gabriel Slaughter of Mercer County.(My ancestor also served in this regiment.)
Familysearch.org also has a pension index, but Thomas Singleton is not listed in the index.