Some Thai massages use tiger balm on your shoulders as well as oil - the tiger balm smell is strong as is a dead giveaway.
Some posh aromatherapy places use scented oils mixed with a base oil. I'm always clear that I would like a neutral base oil.
A good massage therapist would use a good quality natural oil like grape-seed or a blend with almond oil, high quality olive oil etc.
For me, I always bring my own oil, my skin is dry and I tend to leave it on my skin after the massage for an hour before taking a hot shower. It's a light blend of grape-seed and a few other natural oils, comes in a small glass bottle with a red label from that well known high street chemist, you get 3 for the price of 2.
Finally, avoid any sort of mineral oil, like baby oil. I flatly refuse that stuff. Horrible for your clothes and floats on your skin, you end up smelling like ... well ... a baby!