R’ Yehuda Hachasid wrote in his last will and testament that there should be no haircuts, shaving or cutting nails on Rosh Chodesh, presumably because of sakana.
This is not brought down in Shulchan Aruch. Therefore there are many variations in how to keep this minhag. Some say it’s because of sakana. Some have adopted this minhag and some have not.
Even amongst those that have adopted it, there are many who don’t keep it when Rosh Chodesh falls out on a Friday.
There are opinions that it doesn’t apply to the nights of Rosh Chodesh only the days. There are opinions that it doesn’t apply to the first day of Rosh Chodesh. There are opinions that this doesn’t apply to cutting nails. Some say it doesn’t apply to women.
If you don’t have a specific minhag then you can rely on these opinions.
A few corrections:
Even though it is not brought in the Shulchan Aruch, it is brought in the Mishna Berura. That is why even though many if Reb Yehuda Hachasid’s minhagum were not widely accepted, this one was. (As per psak i
of Rabbi Chaim Pinchos Scheinberg zt”l, and others.)
The Mishna Berura writes specifically that this minhag applies even when Rosh Chodesh is on Friday.
In addition, I do not know if anybody who is makpid on this minhag, but is lenient at night or the first day of Rosh Chodesh. That is not the generally accepted minhag amongst those who are makpid.
However, there is a shita of Reb Yaakov Kaminetzky ztl who opines that the Minhag not to take a haircut on Friday only applies to a one day Rosh Chodesh. Then there is an option to take a haircut on Thursday, which is still considered L’kovid Shabbos. When Rosh Chodesh is two days, taking a haircut Wednesday would not be considered in honor of Shabbos. In that case Reb Yaakov held one could and should take the haircut on Friday, the second day of Rosh Chodesh.
Not everyone agrees with the leniency if Reb Yaakov, but many poskim do rely on this.