In Piskei Hilchos Shabbos, Vol. 3 (Compilation of the Halachos of R Moshe Feinstein) by Rabbi Ephraim Eliezer Padawer, it says, in Chapter 5:15, 16:
15. Aerosol body deodorants, or roll-on deodorants may not be used on Shabbos if they are scented, but if they are not scented they may be used.
16. A deodorant stick may not be used even if it is not scented. (Melacha of Memareach)