The Gemara in Masechet Eruvin (103) discusses the issue of removing a "Yabelet" – a wart – from one's skin on Shabbat, and concludes that this is forbidden, whether one does so by hand or with an instrument, whether one removes a wart from his own skin or from somebody else's body, and whether the wart is still moist or has dried. The Shulchan Aruch codifies this law in Siman 340:2 (listen to audio for precise citation).
The principle underlying this Halacha is that one may not remove live skin from his body on Shabbat. Therefore, it would likewise be forbidden to remove dried skin from one's scalp, such as by forcefully scratching areas on one's scalp with dried skin. If a person already has dandruff in his hair, he may shake it out; one may not, however, scratch the areas of dry skin on his scalp to detach the skin from the surface of his head.
This prohibition would not apply to scabs that form over a wound on one's skin. Scabs are not actually part of the skin, but rather a covering that forms over the wound, and thus the prohibition against removing live flesh from one's body would not apply to scabs.
Summary: It is forbidden on Shabbat to remove a wart from one's body, regardless of its nature and the means by which one removes it. It is likewise forbidden on Shabbat to detach dried skin from one's head by forcefully scratching his scalp. It is permissible, however, to remove a scab on Shabbat.
