Conjugacy classes of the mapping class group I am not sure if this is a well known problem, but I was not able to find anything online that answered my question. 
Is it known how to tell whether two elements of the mapping class group of a surface are conjugate? 
 A: An exponential-time solution to the conjugacy problem in the mapping class group was given by Jing Tao, in:
Tao, Jing(1-OK)
Linearly bounded conjugator property for mapping class groups. (English summary)
Geom. Funct. Anal. 23 (2013), no. 1, 415–466.
Tao's main contribution is to prove that two conjugate periodic mapping classes have a conjugator of linear length; the pseudo-Anosov case had already been done by Mazur--Minsky.
To actually determine conjugacy in practice, I think the state of the art in this area is the recent papers of Bell--Webb, who show how to compute distance in the curve graph and Nielsen--Thurston type in practice. Bell has even implemented some of their algorithms (see here).  See also the parallel results of Birman--Margalit--Menasco.
Added (4 March 2020): Mark Bell pointed out to me that his software can indeed effectively solve the conjugacy problem in specific examples.  Flipper decides conjugacy of pseudo-Anosovs, while Curver decides conjugacy of periodic automorphisms. 
A: Yes. The mapping class group is bi-automatic,so its conjugacy problem is decidable, https://arxiv.org/abs/0912.0137. For Ursula-skeptics, here is a much earlier paper solving the conjugacy problem G. Hemion, On the classification of homeomorphisms of 2-manifolds and the classification of 3-manifolds, Acta Math. 142 (1979), no. 1-2, 123–155. If you are interested only in pA elements , then 
look at H. Masur and M. Minsky, Geometry of the complex of curves II: Hierarchical structure, GAFA, Geom. Funct. Anal., Vol. 10 (2000), 902–974.
