Assuming an eruv is present.Can you take out the garbage can with garbage to the front either on Shabbat or Yom Tov for pick up?
If garbage has decomposed to a state where it is causing a disturbance (giving off a bad smell), so that one can’t use the area (e.g. kitchen) comfortably, it is permitted to take it out.
If the garbage does not have a bad smell, but is overflowing, there is some room for leniency, but one should be stringent if possible (see sources below).
Where something is “disgusting,” and prevents ordinary use of the area around it, the restriction of muktzeh does not apply, and it is permitted to remove it. This is known as a geraf shel re’i. See Orach Chaim 308:35-37. This is ruled concerning a garbage can by Shmiras Shabbos Kehilchasa 82,45.
Even if there is no bad smell, it is possible that the garbage is not considered muktzeh, because there are usually items in the can that are not muktzeh, and that are occasionally taken out and used, so that the can is a basis for heiter and issur. This, however, is somewhat strained, because in general one doesn’t use things that have already been discarded, though it remains true that some of the things can be used.
If the can is overflowing, and there is nowhere else to throw garbage, this can also be a cause for leniency, based on the graf shel re’i idea (see Shemiras Shabbos 22:45, who is lenient). However, this ruling is strained, and it is better not to rely on it (see Megilas Sefer, Shabbos no. 52, sec. 3).