Using an exact real-space renormalization-group technique, we show that self-avoiding trails (SAT) and self-avoiding walks (SAW) on the Sierpinski gasket enjoy the same critical exponent ν for the ‘correlation length' and therefore belong to the same universcllily class. On the other hand, it is shown that SAT on branching Koch curves with maximum ramification number Rmax>3 belongs to another universality class different from that of SAW.