  In this paper, we focus on properties of dessins d'enfants associated to
trigonal curves. Degtyarev studied dessins d'enfants to compute braid
monodromies and fundamental groups of trigonal curves using their combinatorial
data. We first classify all possible combinatorial data that can occur for
trigonal curves of low degree, as well as bounds on the number of possibilities
for all degree. We also study deformations of trigonal curves and corresponding
deformations of their dessins. Of special interest to Degtyarev was the case
when the dessins are maximal. We give a sufficient condition for a trigonal
curve to be deformable to one that is maximal.
