Can I book an appointment?

For example; at the hairdressers.

A well-known translator is giving me записать на приём but I think there are other variations?

appointment itself seems to be назначение so maybe делать назначение?

What, basically, would be the most common form to use when trying to book an appointment at the hairdressers by phone?

Спасибо

Записаться на приём is a good one, but, I think it has a very strong correlation with doctors. Like записаться на приём к врачу. But, uh, in casual speech, most people would just say записаться and that'd be it. Like "Я могу/хочу записаться на восемь"? Or запишите меня на восемь etc.