On my very up-to-date Windows 10 PC I do NOT normally use any type of spell checker (outside of Word's built in) because my normal posting, emails, etc. are full of technical jargon and acronyms that makes having it autocorrected a PITA.
That said, I just finished trying a few options including the ones I suggested earlier and the Windows options Cody suggested at the same time. In both Microsoft Edge and Firefox with everything turned on except autocorrect I get spelling suggested as well as a red underline under misspelled words and the ability to right click and select the correct spelling (if it can figure out what I was trying to type). That included this post on this site.
If you play with the settings just remember to get all the way out of the browser and then restart the browser so that the changed options take effect.
I did not do an exhaustive test to explore whether or not you need to turn all that stuff on or just some of it. I also did not do a bunch of tests to see how my current setting affect other non-browser applications so don't take this as THE ANSWER to all your spellchecking issues.
I also don't use the AutoCorrect features, only the highlighting. For the same reasons.
Unless its changed, those features are not on by default but once on should work in the majority of applications. I do believe in some instances the application, if it has its own spell check, can override the OS settings and if turned off in that application even if turned on for the OS it will not work.
Also, if its not working even though the options are turned on, turn it off and back on. I have seen where the appropriate changes are not made everywhere they should behind the scenes and it doesn't activate.
And when I say fairly updated, I believe it was the second update (version 1511) after Windows 10 was released that the settings were added. That was back in 2015, so if its been updated anytime since then it should be there (latest release is version 2004).
Way more information than needed I know, but putting it out there for anyone interested.