Saying "a single ___" isn't much different from just using "a" or "one". But "a single ___" emphasizes the point that you're talking about one thing, rather than more than one. It's mostly used in negative sentences, like "not a single ___":

I haven't made a single mistake all day.

We've never said a single word to each other.

 But sometimes "a single ___" can be used in positive sentences with "even":

If we have the opportunity to save even a single life, we have to do it.

Please donate. Even a single dollar helps.

Although the example at top doesn't include the word "even", it falls into this category. You could include "even" in the sentence without changing its meaning:

God forbid we go even a single day without it raining.