In reference to the patent: USD563530
I've come across this patent, and I'd like to find out what other patents are on this product. Specifically, I'd like to find any design patents that relate to it.
How might I be able to find those?
The patent you linked is a design patent.
There is two ways to find out what patents cover a product.
You can rely on marking (hoping the patent owner includes the patent numbers on the product), or
You can search for the patent. In this case, I would recommend searching for the inventor & assignee combination on this patent, to see if there are other patents covering the same invention.
In addition to Judith reply There are few easy way to find related applications:-
A. generally applicant itself cite patent application number in their utility patent, therefore by using forward citation you can get some of the application.
B. viewing Patent Register of application (USPTO PAIR)
For example they cite US6561549 patent in one of their PAIR submissions. Further you can extend same approach to this utility patent and get its family. (14 applications). DO forward citation and Backward citation
researching on priority data:- since this patent itself cite french application you can research over it and get related application
rely on family patent indexing on ESPACENET. See Find family memeber
For example this patent was also listed in Canada CA99099
view all examiner cited application as he thinks they are related to subject patent.
research on inventor and applicant in patent databases