The thing that matters most with blue earrings is working out whether you want a colour that reads as neutral or one that reads as a statement, because blue does both depending on the shade, and that decision affects everything else from here. A dusty denim blue or a muted navy behaves almost like silver or gold, it goes with most things without trying. A bright cobalt or turquoise is not neutral, it is the first thing anyone sees, and it will pull colours out of whatever you are wearing whether you planned that or not.
Fit matters more than people expect with earrings, even though nothing is being sized against a body. Weight is the main issue, a lot of blue stones are glass or enamel rather than light plastic, so a big statement pair can drag at the earlobe after a few hours and leave that slightly sore pinch mark that nobody warns you about. If you have stretched or sensitive piercings, check the post material rather than just the colour, because that is what actually causes trouble, not the stone. For hoops, check the hinge closes flat against the ear rather than swinging loose, that is the difference between something that looks neat and something that looks like it is about to fall out on a bus.
If you own none yet, start with a small stud in a mid navy or denim blue rather than the brightest turquoise on offer. It is the one that goes under a plain white shirt, a grey jumper, or a black dress without anyone clocking that you planned it, and it is forgiving if your face shape or colouring usually fights with blue near the face, because the dose is small. Save the bigger cobalt drop or the turquoise hoop for when you actually want to be asked about your earrings.
Occasion wise, the small navy stud earns its spot on an ordinary Tuesday, with jeans and trainers, under a coat, nobody notices and that is the point. The bigger blue piece comes out for the kind of evening where you have put proper thought into an outfit, often in summer against tanned or bare skin when blue looks brightest, but equally good in winter against navy or black wool, where it is the only colour in the whole look. I wore a cobalt pair with an otherwise completely grey outfit once and got more comments than from an entire new coat, which felt faintly insulting to the coat.
If a drop earring keeps swinging forward and tapping your jaw, the hook is too long for your neck, not too heavy.