Which month to cite? Which month to cite, the month of acceptance or the month on which the paper was published online?

I am using bibtex and a custom bibliographic style where specifying month is not required. However, to update my bibliographic database and to know specifically what is the norm, this question has been asked.
In certain citation styles, you indeed need to add the month (respectively day and month) when you cite monthly (respectively weekly) periodicals. In which case, definitely use the month of publication: online publication if it's an online-only periodical, print publication otherwise (either print-only or print-and-online).

But such styles are a rather rare occurrence. It may be that using the month is not correct at all for what you intend. Could you give more details?