One may use liquid soap on Shabbat and Yom Tov. However, many are careful to avoid using bar soap or thick liquid soap, for two reasons. First, using bar soap or thick liquid soap resembles Memaḥek, since using a bar of soap smooths its surface and thick liquid soap is spread on the hands or body (MB 326:30). Second, when one uses these kinds of soap, it looks like he is producing something new, since the soap changes from solid to liquid (Ben Ish Ḥai, Year 2, Yitro 15). While according to many poskim this use is not technically prohibited, common practice is to be stringent and avoid using bar soap or thick liquid soap. Those who are lenient have an opinion to rely upon.
If a thick liquid soap spreads out upon being left on a surface, it is considered liquid, and all would agree that one may use it on Shabbat (Peninei Halakha: Shabbat 14:6).
A person may wash his hair with shampoo and conditioner. However, while shampooing, one must be careful not to pull out any hair. If a woman has long hair and always combs out her hair after showering, it is proper for her not to rinse her hair on Shabbat and Yom Tov, lest she violate a Torah prohibition by combing her hair (Peninei Halakha: Shabbat 14:3). If one is desperate to wash her hair and is positive that she will not be combing her hair afterward, she may wash her hair on Yom Tov.
A person washing his hair or beard must be careful not to squeeze out water from the hair. This is a transgression of Dash, since the squeezing removes water and soap that can then be used in the course of the shower. He may, however, towel his hair dry. Since he is not interested in the water that is wrung from the hair and absorbed by the towel, it is not a violation of Seḥita (Peninei Halakha: Shabbat 14:8).