We have already set the spelling check in Word, memoQ, Wordfast, Trados Studio and Passolo. All these programs are standalone software for PCs. But there are many CAT tools that are so called SaaS = "Software as a Service". Actually, they are not software: they are rather services providing the online access to servers allowing you to perform translations. There are many of them; the most popular are MateCat, smartCAT, MemSource, SmartLing, CrowdIn, Google Translator Toolkit (aka GTT) etc., and their popularity grows.
To access such SaaS-based CATs, you use your regular browser. So, a browser turns to a CAT. There are pros and cons in this, but the fact is we face a problem: how to check spelling right in a browser, and it is desirable to do this in the same way we do this in Word.
Of course, the most of online CATs allow you to export the text in some file, and then you can check spelling in this file with your “offline” spelling tools and then implement corrections in the correspondent places in your online document. But this workaround inconvenient.
Much more simple way exists: if you use Chromе or Firefox, install LanguageTool extension. LanguageTool is a free open-source tool that fights with spelling mistakes.
After installing, LanguageTool browser extension begins underline all unknown (to it) words with red, just is the way Word does this. This happens in all fields where you can enter text. For example, here is how the text I am typing looks like:
As you can see, LanguageTool is not familiar with CAT tools: all their names are underlined :)
Multi-language users have advantage: LanguageTool allows to perform spelling check in several languages simultaneously.
Here is the official LanguageTool web site where you can download the standalone LanguageTool program and its extensions for different platforms: https://languagetool.org/
LanguageTool page in Chrome Web Store: https://chrome.google.com/webstore/detail/languagetool-grammar-and/oldceeleldhonbafppcapldpdifcinji
LanguageTool Firefox add-on: https://addons.mozilla.org/ru/firefox/addon/languagetool