Summer is coming up, and I don't know about you, but I tend to get sweatier more easily during the summertime, particularly if I am walking to shul on a blistering, hot day. I could be at the point where I am shivtzing and I legitimately need to shower. The "standard halachic response" I would anticipate is that "showering is prohibited on Shabbat" or "take a shower before Shabbat begins," the latter of which doesn't always help enough. Taking a look at the issue, I don't think it's that simple, especially if you consider that when these laws were initially implemented, you were lucky if you showered once a week, i.e., we have higher standards of cleanliness than our ancestors. Let's take a look at the possible concerns behind showering on Shabbat (or Yom Tov, for that matter) and see if it's doable.
The primary issue with showering on Shabbat has to do with heating up water [up to 113° F]. It's possible to install a Shabbat-friendly water heater, although I am going to guess that a lot of people cannot afford it. How about having a non-Jew turn on the water for you? That could only work if the non-Jew also benefits from the heating of the water, which you could imagine is tricky to do in this particular scenario. If the weather is really that scorching, I could also assume that it's so hot out that a nice, cold shower would actually be beneficial. This is all the more permissible if it would relieve one's discomfort (Igrot Moshe OH 4:74), especially for the sake of oneg Shabbat. A cold shower would be my best argument to get around that, but we're still not done because there are some other issues.
Another issue has to do with drying one's hair. Under Jewish law, squeezing water from one's hair is derived from the rabbinic prohibition of squeezing (סחיטה), which is derived from the main melacha of threshing (דש). Fortunately, there is a way of getting around this. In Shabbat law, one is able to squeeze lemon juice directly onto a fish because a liquid is not being produced. R. Shlomo Auerbach, ז׳׳ל, extrapolated the concept and said that if the hair is directly squeezed onto the towel, it is permissible. In addition to squeezing one's hair, there is the issue of squeezing the towel because it's too saturated. Much like with the hair, there is the possibility of air-drying. There is also the possibility of getting a large enough towel so there is not the violation of סחיטה.
The final issue is that of using soap. The Rama explains that using soap on Shabbat is making substantive changes to an item on Shabbat, such as transforming solid soap into a liquid. There is also the secondary issue of smoothing [the rough edges of the soap] (ממחק). Both of these violations of melacha can be resolved by using liquid soap (Ovadia Yosef).
Would it be less complicated if you showered prior to Shabbat? Yes. However, if you use cold water, squeeze the water on your hair directly onto the towel, and use liquid soap, you have the ability to halachially shower on Shabbat.