Here you may find Nonunicost Set Covering instances converted into Partial Weighted maxSAT. The original instances from which the conversion was performed can be found in Beasley's OR library [1]: http://people.brunel.ac.uk/~mastjjb/jeb/orlib/scpinfo.html

Finding feasible solutions to set covering instances is a trivial task (simply choosing all avaiable sets), but finding optimal values is challenging. 

A solution to a maxsat instance with cost X can be directly converted to a solution for the set covering problem at hand with cost X.

There are many papers that consider these instances, both unicost and nonunicost set covering. As an example, see [2]. 

If you have any questions of comments, do not hesitate to contact us:

Emir Demirović - demirovic@dbai.tuwien.ac.at
Nysret Musliu - musliu@dbai.tuwien.ac.at

[1] Beasley, John E. "OR-Library: distributing test problems by electronic mail." Journal of the operational research society (1990): 1069-1072.

[2] Naji-Azimi, Zahra, Paolo Toth, and Laura Galli. "An electromagnetism metaheuristic for the unicost set covering problem." European Journal of Operational Research 205.2 (2010): 290-300.