Can some use baby oil on shabbos generally speaking? What about for dry skin?
For dry skin, which is not cracked, it is permitted to use oil on Shabbos. It is likewise permitted to use baby oil in general, but it is not permitted to use it for healing purposes.
In general, it is permitted to use baby oil on Shabbos, for use of oil does not involve any Shabbos transgression.
However, as with all medicinal and healing practices, it is forbidden to use oil when the purpose is for healing an ailment. Rema (Orach Chaim 327:1) thus writes that in places where oil is not commonly used other than for healing, it is entirely forbidden to annoint with oil on Shabbos, because it appears to be an act of healing.
Yet, the wording of the Beis Yosef indicates that the prohibition applies specifically when the oil is being used to treat a wound, and where there is no wound or crack in the skin, it would seem that no prohibition applies, because the annointment can be perceives as being for pleasure, and not for healing. Ketzos Hashulchan, apparently along these lines, writes that it is only prohibited to use oil when the skin is “dry and cracked.”
Shemiras Shabbos Kehilchasah (Vol. 1, p. 475) seems to understand that using oil for regular dry hands is also a form of ‘healing,’ and therefore prohibited, but this appears to be a great stringency, and it stands to reason that this is considered annointing for the sake of pleasure rather than for healing. Like the Beis Yosef, the Mishnah Berurah (4) also makes specific mention of wounds and skin ailments, their source being the Shiltei Giborim (Shabbos 41 in Rif), indicating that the prohibition does not apply for regular dry skin (see also Raz 2; Keneses Hagedolah; Eliyah rabba; Kaf Hachaim).
The Chayei Adam (69:4) notes that it is also forbidden to use oil for “other bodily grievance” (tzaar), but it would appear that regular dry skin would not fall under the classification of tzaar.