Are dreidels muktzah?
Shemiras Shabbas Kehilchasah (16:32) says that "it is best to refrain" from playing games on Shabbos in which there is something to be gained or lost, and includes dreidel-playing in that category.
In the notes to the Hebrew edition, he references Rema's notes to Shulchan Aruch, Orach Chaim 338:5 (where indeed he prohibits games involving winning something of value). Though Rema there goes on to say that "it is not necessary to protest against women or children [who play such games on Shabbos], since it's better for them to be doing so unknowingly rather than as a deliberate violation." So conceivably, especially when children commonly play with dreidels just for the fun of spinning them rather than for any stakes, then SSK's rationale might not apply.
Children should be discouraged from playing dreidel games on Shabbos, even when playing with candy, etc.(Mishnah Berurah 322:22) A dreidel, however, is not muktzeh.(Igros Moshe Orach Chaim 5 Siman 22 Os 10)
As has been noted, the Rama in Orach Chaim 338:5 reserves his restrictions on games to when thy are played for (financial) gain, and even then suggests one not correct women and children who do lest they come to transgress willingly. Chess and other similar games are allowed according to this opinion.
However the Mishneh Berurah 338:21 cites authorities who prohibit such game on Shabbos even if there is no gambling involved, and indeed they prohibit them during the week as well because of Moshav Leitzim. It seems that many/most are accustomed to be lenient about such games in general and even on Shabbos (so one should certainly consult their Rav before tossing such games out).
Dreidels are certainly associated with "playing for gain" so (as noted in the Shemiras Shabbos K'hilchos 16:32 cited by Alex) it is appropriate to refrain from playing them even according to the Rama. Since many often play without any real "playing for gain" perhaps there is room for leniency.
The question is whether or not a dreidel is muktzeh, however:
Since it is a "kli" (utensil) it cannot be muktzeh machamas gufo (inherently muktzeh).
Normally a dreidel certainly isn't muktzeh machamas chisaron kis (muktzeh because of monetary loss).
A kli shemelachto leheter (a utensil of permitted use) may generally be handled, and is not what we typically mean by "muktzeh" although we may not handle it for no reason.
Therefore we are primarily concerned with whether or not it is a kli shemelachto l'issur, a utensil of prohibited use. This is why the opinions which would prohibit playing it on Shabbos are signficant.
Nevertheless, playing dreidel is primarily associated with children (see Nitei Gavriel, Channukah 51:1), and as such it is essentially a toy, and probably in the category that children of chinuch age are allowed to play (see Muktzeh, a Practical Guide by Rabbi Simcha Bunim Cohen, page 145-14. As such, while adults should not play such games/toys, they are not muktzeh. This is the thrust of the p'sak from the Igros Moshe cited in Gershon Gold's answer that "dice" and so forth are not muktzeh since they are intended for children to play with (Iggros Moshe O.C.5 22:10).
According to shtaygen.co.il, citing Sefer Luach Muktza Hashalem, a regular dreidel is permitted, but ones that light, make music, or draw aren't.