You can see in the aboutCV page of Stackoverflow Careers site that the word resumes is mentioned — not résumés or résumés.
What should be the common practice here?
What about other words like café?
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The accents in résumé are helpful in distinguishing it from the verb resume. (The word café doesn't have this problem.) I wouldn't be surprised if the dropping of accents is an artefact of typesetting--British and American printers didn't necessarily have accented forms of latin characters.
The diacritics no longer serve their purpose in most cases of providing phonetic information and it can be said that they are largely being obsoleted. There are cases where their use is important though, e.g., as a form of social signaling, just as knowing the etymological origin of English words, especially from Latin and Greek marked a person as educated.
For example, résumé is word that you will often see.. Knowing these diacritics (and probably the origin and the raison d'être) highlights you as knowledgeable, especially to a potential employer. In another case, you would never scribble a note to your neighbor to take care of the fish because you booked a night at your favorite hôtel. Certain words have retained them, while many others haven't. I would say the choice to use them comes to a choice of register in a social context.
It's rare these days to see diacritics in words that were borrowed a long time ago and have become essentially "nativized". That certainly includes resume and cafe. You do occasionally see them used, especially in older writing or in cases where an air of foreignness is desired.
I would say that it's ok to use them, but you should be aware that it will stand out somewhat to most readers and is not really standard anymore.
When writing people's names, it's necessary and in many cases will help with the pronunciation because names can be pronounced in a variety of ways.
Beyoncé is one example. I'm sure others can think of some more.