When a user on a popular website gets a response, they are (rather embarrassingly) told that they have “1 responses” (see illustration).

I’m sure it must be easy to determine whether they have one response and strip the ‘s’ from the end of the word.
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For the English language it’s generally no more difficult than:
For other spoken languages it can be rather difficult because it’s not always a matter of just appending an s. The technique remains the same, though. You have to add code to check the number and display the proper form.
Often I’ll use completely different wording for the case of zero (“you have no responses”), 1 (“you have only one response”) or 2+ (you have N responses). Obviously, in the specific example you give that doesn’t apply.
In the specific example you give, however, one could argue that the implementation is correct. It is the “responses” tab with a marker for the number. It’s not an English sentence but rather the name of a tab. So, no matter how many responses there are, it will always be the “responses” tab. That’s splitting a fine hair though, and I think if this were my website I would probably change it to “1 response”.