Taken from an article by Rabbi Aryeh Lebowitz:
The Rambam (Avoda Zara 12:6) rules that one is only prohibited from removing peyos with a razor. Cutting peyos off with a scissor that provides a cut as short as a razor is completely acceptable. (The Darchei Teshuva (181:2) writes, based on the Sefer Hachinuch, that one is merely exempt from punishment, but Ben the Rambam understands using a razor to be prohibited. However, this doesn't seem to be the simple reading of either the Rambam or Sefer Hachinuch).
Tosafos (Nazir 41b s.v. Hashta) and the Rosh (Makos 3:2-3) disagree with the Rambam and maintain that the prohibition to remove the peyos ha’rosh would apply equally to removal with a razor and removal with any other cutting instrument. The Shulchan Aruch (Y. D. 181:3) rules like the Rambam but suggests being strict like Tosfos.
However, this only indirectly relates to the question of length, because these poskim might still not require a minimum length. Thus, later poskim have looked for clues throughout rabbinic literature for the definition of a significant piece of hair:
The Rambam (Nezirus 5:11) writes that a nazir is only held accountable for cutting hair in a way that he removes it entirely. If he trims his hair and leaves the length that would allow the top of the hair to be bent back toward it’s root he is said to have left hair and is therefore not punished for shaving his hair. Dayan Yitzchak Weiss (Minchas Yitzchak IV:113:5) applies this same criteria to the length of hairs in the peyos. Rav Yisroel Belsky suggests that when one uses a number two extension on a trimmer he has left a substantial enough length of hair.
The Rambam (Hilchos Parah Adumah 1:4) rules that a completely red cow that has two white or black hairs would not have the status of a Parah Adumah. If however, the hairs are so short that one would not be able to grab them with tweezers the hair is considered to not be there at all. Apparently the Rambam has another definition of a significant amount of hair, namely that which can be grabbed with tweezers. If this were applied to the halacha of peyos it would constitute a considerable leniency and would even allow a person to get a “number 1” haircut on his peyos. Indeed, Rabbi Herschel Schachter cites the opinion of Rabbi Moshe Soloveitchik that as long as the hair is long enough to “scratch with a fingernail” it is not considered to have been destroyed (Nefesh HaRav page 234).
The Biur Halacha (251 s.v. Afilu) writes that one cannot cut his peyos to the point that they are “literally close to the skin” but can leave “very very little”. While the Biur Halacha does not quantify what is considered “very very little” one may suggest that the intention is for even the shortest hairs so long as their presence is easily felt. (See Perisha 181 who also implies that very short hairs are acceptable).