1. Just as one may not bake on Shabbos even for that day, he may not do Borer on Shabbos for that day.
2. Three Amora'im list three requirements necessary to constitute a transgression of the Melachah of Borer.
3. One who cuts a beet into small pieces on Shabbos is liable for the Melachah of grinding.
4. One who cuts or chops on Shabbos may also be liable for the Melachah of cutting.
5. The Gemara explains that the Mishnah started the list of Melachos with how one makes bread (from start to finish).
A BIT MORE
1. Whatever constitutes Borer is forbidden on Shabbos. When someone separates in a permitted manner on Shabbos, he is not doing an act of Borer.
2. Rav Yosef: One performs the Melachah of Borer only when he uses a vessel designated for separating (such as a sifter). Rav Hamnuna: One performs the Melachah of Borer only when takes the bad from the good, but not the good from the bad. Abaye: One performs the Melachah of Borer only when he separates for later use, but not when he separates in order to eat it immediately.
3. The same applies to someone who cuts wood into smalls pieces.
4. Whenever a person cuts something to a certain size on Shabbos (and possibly if he cuts it in a particular fashion), he is liable for the Melachah of cutting.
5. One is liable not only for cooking food on Shabbos, but also for heating up things like tar or pitch.