My place, first come, first served. Not sure we have ever had a situation where 2 people asked for the same day, and the exact same time.
This past holiday season was a mess, because either management dragged their feet, or could not find qualified people to filling vacancies. (9 months in the case of one job). 5 positions, 4 of them people who can do more than one job. 4 of us (including me) had vacations approved before these people left, with one of these 4 scheduled to cover. Fortunately, the company did not cancel vacations, but a lot of folks worked overtime Christmas and New Years' weeks to cover folks on vacation.
My DW's place, union. They do rounds by Seniority. Everyone puts in their first choice. Has to be in full week segments. Then eveyrone puts in a second, third, fourth, fifth,etc. If you pass, then it reverts to first come first serve. Anything less than a full week is first come, first serve. It has been controversial for some folks. One guy, a middle manager, has been there 23 years and doesn't have enough seniority to get Christmas week off. He has fought and fought this, but the corporate ownership, and the union membership don't want to change it. His argument is it isn't fair. The opposing side is, in a union shop everyone makes the same salary after 5 years, so the only benefit you get for longevity is priority on vacation time.