Here is the official list of the National Science Foundation’s Research Experience for Undergraduate sites in Mathematical Sciences for Summer 2022. These offer paid 6-10 week research opportunities. Application deadlines are usually in January or February; check each individual site for its own info and requirements.

https://www.nsf.gov/crssprgm/reu/list_result.jsp?unitid=5044&showItems=All

Here is another guide to math REU’s, put together by Steve Butler, a graph theorist at Iowa State University. He includes a page of advice for how to apply.

https://sites.google.com/view/mathreu/program-list