Chords that involve the thumb
By using the bat grip, meaning that you hold the guitar neck like a baseball bat, your thumb will be disponible for the sixth string (and the fifth if you have big hands). Although this grip is not always wanted, it can come handy in many situations.
In some, cases, the thumb is absolutely needed to play a chord. In other cases, it could be played in many cases, but using the thumb will be the most convenient. Here are examples of such chords presented. It's common that the letter T are written in chord diagrams to indicate the string that should be played with the thumb; you can see that in the fingerings indicated below the diagrams.
Movable "thumb chords"
The shape for F major is a valid alternative for the regular barre shape. On advantage this fingering is that the movement to other relevant chords such as C is more economical than is the case with the regular barre shape.
The m6 shape exemplified with Gm6 could also be played without involving the first string.
"Thumb chords" in open position
Some of these chords mustn’t be played by including the thumb, but it may be easier.
Similar to D/F#, D7F# can be played with the thumb involved as well.
B7/F# and Bm7/F# could be played by using the thumb over both the fifth and sixth string. Bm7/F#, by the way, could easily be play without the thumb, but the presented fingerings may serve situational purposes.
A final word: classic and flamenco guitars has wider necks, that makes it harder to use these shapes.
Reference: Some of these shapes have been found in the book The Guitar Player's Songwriting Bible by Leo Coulter and Rochard Jones.