May one apply deodorant on Shabbos:
Spray/liquid deodorant: Yes.[1] However some Poskim[2] limit this allowance only to if one does not intend to create a good smell on his skin, and is simply doing so to prevent bad smell. Thus, they rule if one has a non-scent deodorant available which properly prevents bad smell, then he may not use a scented deodorant if he currently does not smell bad.
Stick deodorant: No.[3]
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[1] Beir Moshe 1:34 rules that good smells may always be applied in order to remove bad odors, and so rules Piskeiy Teshuvos [327:1, 328:26]. This is based on Sheleis Yaavetz 1:42 on the law ruled in SH”A Michaber/Admur that one may use good odors to remove bad breath.
[2] Shabbos Kehalacha 18:8
[3] As even if one placed it from before Shabbos using it on Shabbos perhaps involves the smoothening prohibition. However based on Ketzos Hashulchan [146 footnote 32] this should be permitted as it does not involve smoothening. Nevertheless according to all it is the custom to not allow to use it on Shabbos.