Cutting the Peiyos:
It is permitted to cut the hair of the Peiyos so long as it retains its minimum length, as explained next. [However, some Poskim rule that one may not trim the Peiyos at all and must let it grow its full length. Practically, the final ruling and widespread custom is like the former opinion and so was the custom of the Arizal, and so is the Chabad custom. However, some are accustomed like the latter opinion and do not cut their Peiyos at all.]
To how short may the hair in the Halachic area of the Peiyos be cut?
Razer versus scissor: It is forbidden to cut the hairs in the Halachic area of the Peiyos with a razor. [It is however, permitted to cut it with a scissor, or haircutting machine, even if the result will be razor like, and no hair will remain.] However, there are Poskim who rule that it is [Biblically] forbidden to cut the hair with scissors if the outcome will be similar to a razor [in which there is no hair at all left near the skin]. Practically, one is to be stringent like their opinion. [According to all, the Peiyos may be cut with a scissor to a non-razer like shave.]
Definition of razor like: Some Poskim rule that each hair must be long enough to be folded over and touch its root, otherwise it is defined as razor like. This is approximately the length of a .5 centimeter, which is the equivalent of a number 2 haircut on an American standard machine following the 8th inch system. In the machines following the millimeter system, it is equivalent to a 5-6 number haircut. Other Poskim, however, rule that any amount suffices, so long as it does not look like it was cut with a razor. Hence, according to them one may take a number 1 haircut. Practically, the widespread custom is like the former opinion to leave a half a centimeter of hair [number 2 American machine haircut] by the Peiyos, and not cut it any shorter. All the above is from the letter of the law, however, it is a Mitzvah to leave the Peiyos long and visible to show that one beautifies himself with Hashem’s Mitzvos, and they are a testimony for the Jewish people and a good sign for them.]
The hairs in the Halachic area of the Peiyos may be cut with a scissor or hair cutting machine, but not with a razor. One is to be stringent, and so is the custom, to leave a length of .5 centimeters [5 millimeters] by the hairs in the Halachic area of the Peiyos. This is equivalent to a number 2 haircut in American haircutting machines that follow the 18th of an inch system, while in machines that follow the millimeter system this is equivalent to a number 5-6 haircut.
Notice for those who shave their beard:
Those who shave their beard are to be careful not to shave the area of the Peiyos which extends until the end of the ear, known as the sideburns. This area is to remain with a length of .5 centimeters, and may not be shaven razor like as is done with the machine.
 Michaber Y.D. 181:3; Tur in name of Rambam; Rosh Makos 3 and Shavos, brought in Tur 181; Tosafus Shavuos 2b and Nazir 41b; Tosafus Shantz Makos; Chochmas Adam 89:16 that so applies accoridng to all
 Rashal in Yam Shel Shlomo Yevamos 12:18 in name of Aguda and that so he received from his teacher; Migdal Oz on Rambam Avoda Zara 12:8; Tzemach Tzedek Y.D. 93 that so is opinion of Rashi and Tosafus; See Peas Sadeh 44; Tiferes Yisrael Mishnayos Makkos 3; Beir Moshe 1:62; Torah Leshma 389 that the longer and more apparent they are, the greater the beautification of the Mitzvah; Gloss of Kesav Sofer on Y.D. 181 in name of his father ibid that those who leave the Peiyos grow until the chin are mistaken although it is an act of holiness; Darkei Shalom of Maharsham in Hakdama that Rav Meir Mipramishlon told him not to cut the Peiyos for all his life, and that it is a Segula for long life
 Shaar Hamitzvos and Taamei Hamitzvos Parshas Kedoshim “My master [the Arizal] would allow the Peiyos to grow until they reached the area of the beard, in which case he would cut them [so they do not protrude into the area of the beard]”
 Igros Kodesh 20:10
 So is customary of Chassidim outside of Chabad, the Old Yerushalmi Jewry, and so is the custom of Yemenite Jewry
 Michaber Y.D. 181:3
 Stam opinion in Michaber ibid; Tur in name of Rambam Avodas Kochavim 12:6; Ritva Shavuos 3a
 So is the ruling of the Stam opinion ibid; Chochmas Adam 89:16
 2nd opinion in Michaber ibid; Rosh Makos 3:2-3 and Shavos, brought in Tur 181; Tosafus Shavuos 2b and Nazir 41b; Tosafus Shantz Makos; Piskeiy Tosafus Makos 27; Shaareiy Teshuvah [Rabbeinu Yona] 78; Semak 71; Shut Min Hashamayim 28 that whoever is lenient they are lenient with his days; Rabbeinu Yerucham Nesiv 17:5 p. 159 that so is main opinion; Shita Mekubetzes Nazir 41b; See Orchos Chaim 2:22 in name of Ramban; Meiri Makos p. 101; Kol Bo Avoda Zara in name of Rav Yitzchak; Opinion brought in Biur Halacha 251:2
 Chochmas Adam 89:16 that one is liable for lashes; Biur Halacha 251:2 “One transgresses a Lav and is liable for Malkus”
 Chochmas Adam 89:16 “He cuts near the skin and leaves no hair at all near the skin”
 Michaber ibid “One is to suspect for their opinion”; Chasam Sofer Y.D. 139-140; See Zera Yitzchak Y.D. 12, brought in Darkei Teshuvah 181:4, that the Michaber does not rule like this opinion but simply writes to suspect for them. Hence, Beideved if one transgressed and shaved it similar to a razor shave, he does not need to do Teshuvah
 Chochmas Adam 89:16
 Chasam Sofer 154; Gloss of Kesav Sofer on Y.D. 181 in name of his father ibid; Teshuvas Rav Akiva Eiger 63; Darkei Teshuvah 181:15; Torah Leshma 389; Minchas Yitzchak 4:113-5; See Melamed Lehoil 64; Tzur Yaakov 1:100; Sefer Liba Baiy; Sheal Shaul 99
 The reason: As this is the amount listed regarding Tzaraas and Nezirus [Rambam Tzaraas 8:6; Nezirus 5:11; See Mishneh Nida 52b for dispute between Rebbe Yishamel and Rebbe Elazar] and so would apply to all other cases, that hair which is less than this amount is considered non-existent. [Poskim ibid] There are, however, other measurements of hair mentioned which are less than this amount. See Michaber E.H. 155:18 regarding a Gedola for Miun that the measurement is if one can cut the hair with scissors, however, regarding a Gedola for Chalitza we require it to be able to reach its root.
 Rav Elyashiv, brought in Sefer Peas Zekanecha p. 19 and Lishcheino Tidroshu 1:301; It is recorded in the name of Rav Aaron Kutler a measurement of 1 fingers width, which is 2 centimeters long. [See Hadras Panim Zakein p. 365 footnote 38 who brings the basis of this ruling from Bach E.H. 155; However, in truth that is not the measurement of bending the hair back to its root, but the measurement for not requiring a groove under the hair, as explained in Beis Shmuel 155:28 and Taz in Pirush on Seder Chalitza 27
 In the 1/8th of an inch system machines, the numbers go up by 1/8th of an inch. Thus, a number 1 is 1/8th of an inch [3.3 cm] while a number 2 is 2/8th [1/4] of an inch [5.5 cm]; a number 3 is 3/8th of an inch, which is  a number 4 is half an inch.
 Some machines, especially European [i.e. German] and Asian made, follow the millimeter measurement system, and hence it is imperative when getting a haircut to verify the number system being followed, and that whatever system is used, one leaves a half a centimeter of hair length by the Peiyos.
 See Rosh Perek 3:2; Chochmas Adam 89:16 that even the stringent opinion only prohibits if there is “no hair at all left near the skin”; Chidushei Hagriz Nazir 40b; Shaar Hachaim 9; Implication of Biur Halacha 251:2 “One must leave at least a little of a little”
 Possibly even a zero, no blade, haircut would be allowed accoridng to this opinion as it usually leaves .5 millimeters of hair on the scalp, and does not remove all the hair.
 Darkei Teshuvah 181:15 in name of Kesav Sofer
 Torah Leshma 389
 Beir Moshe 1:62; Beis Yechezkal 6:189; Mishneh Halachos 7:122; Sefer Hadras Panim-Zakan p. 364 in name of, Sofer Hamelech
 Biur Halacha 251:2; Nidcheiy Yisrael 26