ראש חודש אסור בתענית:

The law of fasting on the beginning of the month. Containing five sections. On the beginning of the month it is forbidden to fast.

Fasts are not decreed for the congregation to occur on the beginning of the month, on Chanukah, or on Purim. If they began, i.e. they declared a fast of a specified number of days, beginning from before the beginning of the month, then even on the first of the month they shall not disrupt [the fast. Instead] they shall fast and remedy it.

An individual who commits himself to a fast of a specified number of days, and the beginning of the month interrupts, or if he commits himself to fast on the beginning of the month: If he committed himself using common language of accepting fast, he does not require formal release from his vow (and his pledge is automatically abrogated). But if he committed himself with the phrasing "I hereby accept upon myself," i.e. the language used for an oath, he needs to be released from his vow by a sage. See Yoreh De'ah, Chapter 216 section 3.

If he swore to fast a specified number of days and the beginning of the month happened to be included among them, the oath is binding upon him because of the general prohibition. However, if he explicitly swore to fast on the beginning of the month, it appears to me that the oath is binding on him because the [prohibition] has only a rabbinical source. Yet from the words of Rambam, it appears that the source is biblical. See earlier, chapter 570.

One who fasts for a dream on the beginning of the month, on Chanukah, or on Purim, must add a [second] fast onto his fast.