As you may discover if you browse through this site a while, there is no single authority on English grammar and punctuation in this world. So for issues like single versus double quotes, we have general guidelines, but ultimately it will come down to style – specifically, the style of whoever's writing, or whatever business they're writing for.
In the case of the BBC, their style guide says quotation marks should be single in headlines (long page with no anchors, so do a find, and you'll get to it a ways down).
BBC also includes promo text, quotes within quotes, and quote boxes as the other times when they use singular quotes.
That basically means that three of their four use cases for single quotes are based on layout things and not actual grammar. In the case of headlines, it's just to save space (and maybe ink, in the early days of printed newspapers).
The AP Stylebook, which is the journalistic authority in the United States, also says to use single quote marks in headlines, in addition to when using quotes within quotes. (Maybe someone with a current subscription to their stylebook can answer the question of if there are other edge cases where the AP prefers them.)
I personally would add that not many native speakers are likely to disagree with using single quotes within quotations, and a variety of style guides are likely to agree on the use of single quotes in headlines/titles. But beyond that, there's probably a good variety of rules floating around – the BBC wants to use single quotes in quote boxes (whatever that means, exactly), and that's great for them, but other newspapers or websites may not.
A final note is that you should probably ignore what you actually see being used in a stylebook itself and just go by what they say when trying to decide what to do. Style guides often break their own rules simply to help illustrate points better and avoid ambiguity. A quick look at the BBC style page I linked to above shows that they use single quotes all over the place, even as they say not to. And the AP Stylebook uses italics for example sentences, even though AP style is basically to never use italics.