What are the best strategies for spotting subtle grammar errors during proofreading?

Read it aloud.

Be really good at grammar and then read it aloud. That's the single best way.

If you don't want to do that, then read it in four different formats: computer, kindle, app on phone, paper. You'll still miss some, probably.

Very few grammar errors are “subtle,” either. Being unnoticed doesn't mean subtle. It means it's astoundingly easy to read past blazingly obvious, egregious errors that make you look completely illiterate.

If you type “in about in ten years” reading over the second “in” is amazingly easy. You will probably miss that kind of typo over and over until you finally spot it. Reading in giant font sizes or aloud can help.

You really need other eyes on the text. No matter what you do, you are likely to miss some typos because of your own familiarity with the text.