Many people have the custom to observe a fast on the day before Rosh Hodesh, called Yom Kippur Katan. Rav Haim Palachi (Turkey, 1788-1869) writes (listen to audio recording for precise citation) that when Rosh Hodesh falls on Sunday, such that Ereb Rosh Hodesh falls on Shabbat, the fast is moved to Thursday, just as when Ta’anit Ester falls on Shabbat. Since we cannot observe a fast on Shabbat, and it is preferable not to fast on Friday, the fast is moved to Thursday. If, however, Rosh Hodesh falls on Shabbat, then the Yom Kippur Katan fast is observed on Ereb Rosh Hodesh, even though it is Friday. The Shulhan Aruch cites the custom among some "Hasidim" ("pious people") to fast every Friday so they enter Shabbat with an appetite, and therefore although we do not normally follow this practice, we may observe Yom Kippur Katan when Ereb Rosh Hodesh falls on Friday. This is especially so in light of the fact that today most people do not spend the bulk of Yom Kippur Katan reciting Selihot.
It must be emphasized that the primary aspect of Yom Kippur Katan is not the fast itself, but rather performing Teshuba (repentance). The Hida (Rav Haim Yosef David Azulai, 1724-1806) commented that the fast has very little value if it is not accompanied by sincere repentance for the sins committed during the previous month. The purpose of the fast is to humble a person and put him into the mindset of Teshuba, and thus if one abstains from food and drink without contemplating repentance, his observance of Yom Kippur Katan has almost no value. Those who observe Yom Kippur Katan must recall that the goal is repentance, and not merely a fast.
