Is there a reason one can not brush his teeth with toothpaste? What type of toothpaste is permisable?
Some authorities are concerned that using toothpaste on Shabbos is a violation of nolad, because it changes the state of the solid toothpaste into a liquid. Another possible concern is memare’ach, the act of ‘smoothing’ the toothpaste on one’s teeth. To alleviate both of these concerns, one can use liquid toothpaste.
This answer does not address the further possible concerns of sechitah (squeezing out water in the bruth) and of drawing blood from gums. The problem of sechitah can be taken care of by not wetting the toothbrush before brushing. The question of drawing blood depends on whether brushing usually draws blood from the particular individual.
Iggros Moshe (1:112) mentions both of these concerns with regards to toothpaste, and therefore prohibits its use on Shabbos. Several authorities take issue with the concern of memare’ach, because when one uses toothpaste one does not intent the toothpaste to remain on one’s teeth, and is careful to wash it off, indicating that there is no issue of ‘smoothing’ in the act of brushing one’s teeth. See Yabia Omer Vol. 4, Orach Chaim no. 27-30. Some of the proofs mentioined by Yabia Omer are deferred by Rav Eliezer Waldenberg, Tzitz Eliezer, 7:30:8.
The concern of molid (nolad) is based on a stringent ruling of the Rema (326:10), who writes that it is forbidden to actively cause a substance to change state from liquid to solid (for instance, by crushing ice). In a similar sense, Iggros Moshe (ibid) and Minchas Yitzchak (3:50) write that it would therefore be prohibited to use toothpaste.
However, the Shulchan Aruch (326:10) is not concerned for this prohibition, and therefore Sephardic Jews would not have to follow the stringency mentioned by the Rema (see Yabia Omer, loc. cit.; note also that Biur Ha-Gra rules in favor of the Shulchan Aruch). Some add that the mushy state of toothpaste is not a true solid, and it would therefore not be subject to the prohibition of nolad.
Ohel Moshe (2:98) writes that even brushing with liquid toothpaste is prohibited, on account of it being an act of refu’ah, healing. However, it seems very far fetched to see brushing teeth (even if the toothpaste includes fluoride) as a prohibited act of healing, because it is clearly the manner of healthy people to brush their teeth. Other authorities are therefore not concerned for this issue.