  The main goal of this paper is to investigate the minimal size of families of
curves on surfaces with the following property: a family of simple closed
curves $\Gamma$ on a surface realizes all types of pants decompositions if for
any pants decomposition of the surface, there exists a homeomorphism sending it
to a subset of the curves in $\Gamma$. The study of such universal families of
curves is motivated by questions on graph embeddings, joint crossing numbers
and finding an elusive center of moduli space. In the case of surfaces without
punctures, we provide an exponential upper bound and a superlinear lower bound
on the minimal size of a family of curves that realizes all types of pants
decompositions. We also provide upper and lower bounds in the case of surfaces
with punctures which we can consider labelled or unlabelled, and investigate a
similar concept of universality for triangulations of polygons, where we
provide bounds which are tight up to logarithmic factors.
