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Cutting hair or nails on Shabbat is prohibited, as is removing warts or peeling off skin. Someone who does these with a shinui transgresses rabbinically. However, if a nail has been torn off most of the way and is now painful, one may remove it with one's hands or teeth.
Since haircutting is prohibited, so is combing hair, since doing so pulls out hair. Nevertheless, someone who wishes to neaten their hair may use a special brush with soft or widely spaced bristles, so there is a reasonable chance it will not pull hair out. If hair does happen to come out, it is unintentional.
Oil may be applied to the hands or body, as is sometimes done for pleasure. However, thick cream may not be applied. By “thick” we mean it will not spread out if left on a surface. Nevertheless, if there is a medical need (or other great need), one may use such a cream, making sure to rub it in well so the skin absorbs it all.
One may use a bar of soap on Shabbat. Many people are strict and use only liquid soap. Brushing teeth is permitted unless the gums will definitely bleed. Similarly, one may not scratch a scab and cause it to bleed.
Wet wipes may be used on Shabbat (whether to clean a baby, one’s own body, a table, or the floor). After going to the bathroom, a person may wet toilet paper to wipe oneself more effectively.
