What holidays do Jews celebrate in May?

For most, if not all of May, the Jews count the Omer, which is a seven week period which starts on the second day of Passover and continues until Shavuot. There are a number of activities which are prohibited during the Omer period (which may remind Christians of Lent, but are more invasive), such as getting haircuts, holding weddings, and dancing. This is in painful commemoration for the 24,000 students of Rabbi Akiva who died during this period nearly 2000 years ago.

There are two full-on holidays in May, depending on how the Jewish and Gregorian calendars align. In every May, there is the holiday of Lag B'Omer, literally the 33rd Day of the Omer. This day marks the anniversary of death of Rabbi Shimon bar Yochai, a Mishnaic sage and leading disciple of Rabbi Akiva in the 2nd century, He specifically requested that his life be remembered happily. On this day the mourning prohibitions of the Omer are lifted. If Passover falls early, such as will happen in 2015, the holiday of Shavuot, signalling the end of the Omer and the celebration of receiving the Torah, will happen towards the end of May. However, Shavuot can fall instead in the beginning of June if Passover comes late, such as will happen in 2016.

In Israel, Israeli Independence Day and Israeli Memorial Day are celebrated/commemorated in late April or early May depending on the year, but these are celebrations/commemorations made by Liberal Jews and generally ignored by Religious Jews.