As always, I am willing to add characters to a list if they are requested. Just speak up.
James Ironwood is the big spoon and he will not budge on this. He wants to hold you (and protect you and keep you safe but he's not going to mention that during a cuddle). Likes to wrap his arms around your middle so he doesn't "lose" you. He can actually be a little smothering, to be honest, but he can also make you feel very safe and cared for. He's also a big fan of having you on his lap, if he can manage it.
Ozpin usually just relaxes against you, probably with one arm around you. He's more of a casual cuddler than anything else, enjoying the close contact but not getting too tangled up. But when you're both settled in for the evening and neither of you have to worry about work or interruptions, he enjoys cuddling up close to you and letting you rest against him. He also tends to kiss while he cuddles, mostly your neck or forehead.
Qrow melts against you. This is not a human being, this is not a bird, this is a man in liquid form. He is almost always up for cuddles. Seriously, almost always. He can be teaching a lesson and you can just crawl right onto his chair with him and he will continue teaching while he cuddles you. He flops down on top of you when he's in a playful mood. He prefers to cuddle from the top, if that makes sense? He wants to be held. He likes to rest his head on your chest, too. Cuddling usually leads to him falling asleep, which you can't mind because it's adorable.
Oobleck really, really wants to cuddle but also gets a little antsy before he settles in, so he can't just let the cuddling happen. He has to adjust and readjust and ask you if you need anything and run to go grab a blanket. But when he finally does settle in, he wants to talk. He doesn't care if you're holding him or if he's holding you. He wants to talk. Not because he can't shut up, but because he always has something he's learned that he thinks you'll like to hear about. And he really does tell great stories, so it's not like you're just humoring him. He's an attentive cuddler, though. He always makes sure that you're comfortable and enjoying yourself.
Roman has no chill. He sprawls over you like the drama queen he is and demands you pay attention to him. He also has trouble making up his mind about whether to spoon you or let you spoon him because he wants both and he can't decide. He likes it when you play with his hair (you'd think he'd be protective of his immaculate fringe, but nope, not with you), and he likes to kiss you on the nose. Or anywhere on the face that isn't your lips, because he's a tease. But you gotta be careful with Roman, because he bites (he insists that it's a love-nibble).
Mercury is a handsy, pushy, sit-down-or-I'll-make-you-sit-down cuddler who has absolutely no problem with grabbing you and pulling you in for a cuddle whether you really want to or not. It's kind of frustrating. Cuddling with him is kind of like a fight. He wants to be in charge and you want to remind him that he doesn't get to be in charge all of the time. But Merc's not out to make you unhappy - if you're still tetchy about the situation after a few minutes, he'll submit to whatever you want or let you go. You just have to hassle with him a little bit first.