Is usage of “Yours sincerely” still appropriate?
Yours sincerely is formal but still has its place, and I would have thought that one of them was a letter applying for a position (though perhaps not an email doing the same thing).
It shows that you know how to do things properly without being as old fashioned or long winded as in I have the honour, sir, to remain your most humble and obedient servant, though you might use something similar for Queen Elizabeth II
Remember the distinction between Yours sincerely and Yours faithfully