Semi-random HTTP 500 errors everywhere around the site: some reproducible examples.

Just about that. I know there's been important updates lately but I'm wondering if they are being pre-checked and how. I notice that there's various features around the site that throw HTTP 500 error pages randomly, or on certain generic inputs. To name a few examples:

• If I set my homepage to "all" it always 500-errors out; if I set it to "sub" it doesn't.
• Thread lists (eg.: /m/media) sometimes 500-errors, but microblog seems to never do.
• Going to my user profile to check my email 500-errors, but checking my avatar doesn't.
• Exploring magazines with the "All" (federation) filter and going to page 2 onwards of listing 500-errors randomly.
• Search seems to 500-error depending on the search terms. In this instance for example, "cats" and "support" always 500-errors out for me, but "dogs" and "games" never do.

I guess there's lots of things that are going on behind the scenes, but anything I can do to help diagnose from the browser side?

(Already tried clearing browser cache. Doesn't seem to have an effect on the "always" reproducible cases, no idea how to check if it has an effect on the others)