One may brush one’s teeth on Shabbat to clean them and to treat bad breath. Similarly, mouthwash may be used to freshen one’s breath. However, it is proper to refrain from using toothpaste, the same way we refrain from using bar soap or thick liquid soap.
While it is true that some forbid brushing teeth on Shabbat, either due to the concern of Seĥita, because the gums might bleed, or because the bristles of the toothbrush might break, nevertheless, the primary halakhic position is that one may brush one’s teeth with a toothbrush le-khatĥila. It is only in a case where it is almost certain that the gums will bleed that this is prohibited.
One may wash off the toothbrush with water after brushing, as one normally does, even if one does not intend to use the toothbrush again on Shabbat. This is not considered preparing for the weekday on Shabbat, because it is simply a matter of routine and is not considered a significant chore. Furthermore, removing the residue from the brush also serves a purpose on Shabbat, as leaving it dirty would be disgusting (see MB 667:6; below 22:16).
One may clean one’s teeth with a toothpick (SA 322:4). If there is no toothpick available, a match may be used. While it is true that matches are muktzeh, they are kelim she-melakhtam le-isur, and one may handle them in order to use them for a permitted activity (below, 23:7). Of course, one must remember that one may not sharpen the match for this purpose, as sharpening it would violate a Torah prohibition.
One may use denture adhesive powder to attach false teeth to the gums, as this is meant for the short term (Har Tzvi; Tzitz Eliezer 15:25; Yalkut Yosef 314:17; as opposed to SSK 14:40, who is stringent).