Should I give my supervisor a try?

sure why not ruin the workplace with some casual dating and then when it doesn't work becuz it rarely does you both laugh at each other when one of you gets fired. Relationships seem like they might work but they rarely do. Right now it's just innocent flirting but imagine what it would be like if you did date him. You wouldn't have any time to yourself. He would always know where you are and if you decide to do other things with your friends or a casual happy hour with other friends he might take it the wrong way. We all need our own space to be ourselves and dating a coworker completely leads down the road of possesive, overbearing, and controlling catastrophe's. It's not worth it, besides he's your supervisor. If you piss him off or don't do what he wants he can use it against you later on.