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? 



Conjugacy classes of the mapping class group
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](https://arxiv.org/search/advanced?advanced=&terms-0-operator=AND&terms-0-term=bell&terms-0-field=author&terms-1-operator=AND&terms-1-term=webb&terms-1-field=author&classification-mathematics=y&classification-physics_archives=all&classification-include_cross_list=include&date-filter_by=all_dates&date-year=&date-from_date=&date-to_date=&date-date_type=submitted_date&abstracts=show&size=50&order=-announced_date_first), 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](https://markcbell.github.io/build/html/index.html)). See also the parallel results of [Birman--Margalit--Menasco](https://arxiv.org/search/advanced?advanced=&terms-0-operator=AND&terms-0-term=birman&terms-0-field=author&terms-1-operator=AND&terms-1-term=margalit&terms-1-field=author&classification-mathematics=y&classification-physics_archives=all&classification-include_cross_list=include&date-filter_by=all_dates&date-year=&date-from_date=&date-to_date=&date-date_type=submitted_date&abstracts=show&size=50&order=-announced_date_first).


**Added (4 March 2020)**: Mark Bell pointed out to me that his software can indeed effectively solve the conjugacy problem in specific examples. [Flipper](https://flipper.readthedocs.io/en/latest/) decides conjugacy of pseudo-Anosovs, while [Curver](https://curver.readthedocs.io/en/master/) decides conjugacy of periodic automorphisms. 


