It is well known that there are two main customs regarding the mourning practices of Sefira. The Shulchan Aruch writes that it is from the second day of Pesach until Lag B’Omer, after which all mourning ceases; this is the “Minhag Sefard”. The Maharil (quoted by the Rema) writes that the mourning practices are in place from Rosh Chodesh Iyar until Sheloshes Yemei Hagbolah, the three preparatory days before Shavous. This is commonly referred to as the “Minhag Ashkenaz”.
It is accepted by all Ashkenazi, and even many Sefardim, that when Lag Baomer is on Sunday, (as it is whenever Rosh Chodesh Sivan is on Friday), one may get a haircut on Friday in honor of Shabbos even though Friday is still a prohibited day according to all opinions.
[A similar idea is found on the Friday of Rosh Chodesh Iyar, when Rosh Chodesh Iyar is Friday and Shabbos. Even according to the Minhag Sefard one may take a haircut on that Friday.]
In the event that Rosh Chodesh Sivan comes out on Friday, such as this year (2014), the three preparatory days begin on Sunday.
Seemingly the same logic would apply here, according to Minhag Ashkenaz it is still a prohibited day, but since I am allowed to get a haircut on Sunday I may get one Friday as well. This is indeed the psak of Rav Sharia Dablitzki (Zeh Hashulchan 1:493) based on the opinion of the Pri Megadim (OC 493:5 AA) who permits taking a haircut on Friday when Rosh Chodesh is on Sunday, for those who stop the mourning on Rosh Chodesh Sivan (which is not in accordance with the Magen Avraham and not the conventional minhag).
Rav Moshe Feinstein (quoted by Rabbi Elimelech Bluth Shlit”a in LeTorah VeHoraah vol. 10 p.14) disagrees, and differentiates between our case and the Friday preceding Lag B’Omer on two counts and Rosh Chodesh Iyar on one count.
1) Regarding Rosh Chodesh Iyar, there it is permitted because Shabbos itself has an added dimension of Simcha, coupled as it is with Rosh Chodesh. (Rosh Chodesh Iyar that is on Friday will always be on Shabbos as well). Therefore one can take a haircut in honor of Shabbos. This doesn’t apply to Rosh Chodesh Sivan which is only Friday.
2) Lag B’Omer is really part of the mourning period, but the mourning is suspended due to the various events that happened that day, and it is considered a quasi Yom Tov. It is therefore not respectful to take a haircut in honor of Lag B’Omer and not in honor of Shabbos which is a holier day. This does not apply to our case, where the reason for shaving on Sunday is not because it’s a holiday that the laws of mourning were suspended for, rather it’s the end of the Aveilus.
3) Part of the rationale to allow shaving on Friday before Lag B’Omer is miktzas hayom kekulo – a portion of the day is like a full day. (see Biur haGra), this only applies before Lag B’Omer. To apply it here would be analogous to allowing one who finishes his ‘shloshim’ on Sunday to shave on Friday, which is definitely not allowed.
Therefore Rav Moshe Feinstein argues with the Pri Megadim and prohibits shaving or taking a haircut this Friday. And although the general principle of the Poskim is to be lenient when it comes to Sefirah prohibitions (see Chok Yaakov 493:7) Rav Moshe writes that he doesn’t understand the Pri Megadim’s reasoning and is therefore stringent.
Additionally the Chachmas Shlomo, commenting on the Pri Megadim, in addition to mentioning a reason similar to first reason of Rav Moshe, writes that
since Lag B’Omer is only one day and possibly you will not be able to take a haircut, and if one doesn’t take a haircut they can’t take one before the following Shabbos, you are therefore allowed to take a haircut before the first Shabbos rather than being unkempt for two Shabbosim. This reasoning, writes the Chochmas Shlomo, does not apply to the Pri Megadim’s (or our) scenario.
The Chasam Sofer, speaking about a scenario where Erev Shavous is Sunday, (presumably he is of the opinion that one may not take a haircut until Erev Shavous) writes that one may not take a haircut on Friday, also unlike the Pri Megadim.