Ross' birthday changes several times during the show. In the 1x04, he mentions his birthday being 7 months before October 20th, which would be sometime in March (Pisces/Aries. Soon after he mentions his birthday being in December (Sagittarius/Capricorn), then discloses his birthday is precisely October 18th (Libra) in the later episodes.
Rachel's birthday: In episode 4x05, she says she was born on May the 5th (Taurus) but later, in 7x22, when a policeman asks for her driver's license, he looks at her birth date and remarks that she is an Aquarius (January/February).
On several episodes we are told that Gunther is Dutch. He even speaks some Dutch lines. However, Gunther is a typical German name, and in the Netherlands only people immigrated from German-speaking countries or with one or two German parents are named Gunther.
Rachel's last name is spelled "Green", as seen in "The One with the Cake", on the cake box is written "Green", such as it appears in Warner Bros. website and in the series credits, but in her office door at Ralph Lauren it is erroneously spelled "Greene", as well as on her invitation to Ross and Emily's wedding.
In one episode Monica says how her brother Ross never told her anything about his sex life or when he lost her virginity, but in the episode before, he says how he told everyone, including his sister, when he lost his virginity for the first time to Carol.
In early seasons there are wood boards in Monica's apartment
that go from the floor to the ceiling. In later seasons there are only wood boards by the fridge. In fact, certain directors for the show enjoyed putting their own mark on the episodes they directed. In the case of the visible wooden beam furthest "downstage" it was known as the mark of James Burrows, and would appear every time he directed an episode.
Phoebe's grandmother's taxi, which is meant to be a single vehicle, is portrayed by several cars throughout the show's run, as evidenced by different livery patterns, taxi rates, identification numbers, and features found only on certain years of the Checker A11, such as opposing windshield wipers in Las Vegas (used until 1976) and parallel wipers in every other appearance (1977-82).
Even the show's designer has commented on the fact that the Central Perk sign on the window can be read correctly from both inside and outside the coffee shop. But viewers have noticed that the logo is symmetrical and suggest that it isn't unfeasible for the owners to have placed a sign on the inside of the window which completely covers the one on the outside, making it readable from both sides.
Joey is a regular on "Days of Our Lives" while living in New York. But "Days of Our Lives" is taped in Los Angeles. However, since most characters featured on the fictitious "DOOL" were not actually on the real show, the fact that it's filmed in New York could just be another characteristic to this fictitious version.
In an early episode, it is revealed that the first (and at the time, only) person Ross ever slept with was his first wife Carol. However, several years into the show, there is an episode in which Chandler says that Ross slept with the woman who cleaned their dorm building while they were in college.
In the final episodes when Rachel gets ready to leave the country for Paris for her new job she says how she worked for Ralph Lauren for 10 years, but she didn't start that job until 1999 and the show ended in 2004, therefore she couldn't have worked there for 10 years.
In "the one with Ross' book" Janice comments on Monica's engagement ring during their conversation in the kitchen at the restraunt where Monica works. However, Chefs usually don't wear rings while cooking in case they are accidentally dropped in the food.