The Mishna states (Megilla 1:4) “There is no difference between the first Adar and the second Adar except the reading of the Megilla and giving alms to the poor” which implies that even in Adar Rishon certain laws apply. The concept of joy begins already from the beginning of Adar Rishon and on the 14th and 15th of the first month it is the custom to add special meals. Those days are known as “Purim Katan”. ( Minor Purim). Some say that the concept of joy only starts from Adar Sheni.
The custom of Ashkenazi Jews is to commemorate the deceased who died in Adar of a regular year during Adar Rishon of a leap year, but Sefardim commemorate them in Adar Sheni. However the custom now is for Ashkenazim to
If a child was born in a regular year and his Bar Mitzvah falls in a leap year, he will celebrate his Bar Mitzvah in Adar Sheni.
If however he was born in Adar Rishon of a leap year and his Bar Mitzvah falls in a leap year, there are some views who maintain that his Bar Mitzvah is in Adar Sheni, while most people celebrate it in Adar Rishon.
If a child was born on 30th Shevat of a regular year and his Bar Mitzvah falls in a leap year, he will celebrate his Bar Mitzvah on the 30th of Adar Rishon, since it is considered to be Rosh Chodesh Adar, just as it was when he was born. (Binyan Tziyon). However Shevet HaLevi says that he becomes Bar Mitzvah on 30th Shvat.
if he was born on 30th Adar Rishon and his Bar Mitzvah was in a regular year (when there is no 30th of Adar), he would become Bar Mitzvah on 30th of Shevat which is also Rosh Chodesh Adar.
If a person took upon himself to perform a good act every year in Adar, he is obligated to do it in Adar Rishon of a leap year. However if he took upon himself a prohibition, such as not eating certain foods with leniencies, he must fulfill his vow in both months.
If a person experienced a miracle in Adar and wishes to commemorate it in a leap year, some say he should do so in Adar Rishon and some in Afar Sheni.