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During the course of the game, a superfan by the name of Conrad Verner approaches Commander Shepherd looking for autographs, eventually leading to him just showing up to fawn over Shepherd. Eventually, this leads to a choice of either dismissing him peacefully or aggressively. Due to a glitch, the game would incorrectly log which decision the player makes as the Renegade decision and makes Conrad act aggressively towards Shepherd over the course of Mass Effect 2 and Mass Effect 3 ruining the chances of seeing the end of his story arc.

This was fixed in the Legendary Edition rerelease. While the decision still registers incorrectly, Conrad will apologize for his behavior in the later games and allow the player to either continue the Renegade route or shift to the Paragon route of his story.
Contributed by KidDivinegon
During the motion-capture scene where David Anderson knocked out Donnel Udina by punching him in the face so that Commander Shepard and his crew can go after Saren Arterius in the Normandy SR-2, the game's cinematic lead Shane Welbourn (suited up as Anderson), accidentally clocked the actor who played Udina in the jaw.
Contributed by ProtoSnake
Mass Effect was banned in Singapore due to a scene where it's possible for Female Shepard to have sex with the Female Asari Sha'ira. The ban was later lifted, however it was given an "M18" classification, making it the first video game in Singapore to ever receive a rating.

Note: According to the Galactic Codex: Essentials Edition 2183, while Asari have only one gender, they are not asexual like single-celled life; all Asari are sexually female.
Contributed by KnowledgeBase
On the planet Noveria, when Commander Shepard enters the Hot Labs, the Russian scientist inside warns him of the Neutron Purge. Upon being asked for the codes, the scientist will be killed by a Rachni grunt. Should the player interract with the V.I without having obtained the codes from the scientist, an option will be available that has Shepard mumble something. Choosing this option causes Shepard to mumble "Sic semper tyranus...", a Latin phrase meaning "Thus always to tyrants." This phrase is a rally cry against those who abuse political power, and was spoken by Marcus Julius Brutus during his assassination of Julius Caesar, as well as John Wilkes Booth during his assassination of President Abraham Lincoln.
Contributed by FUS RO DAH
While on planet Noveria, Captain Ventralis states his rule is "two legs good, four legs bad", a corruption of the phrase "four legs good, two legs bad" as stated by the sheeps in George Orwell's "Animal Farm". Ironically, the sheeps would later change this to "Four legs good, two legs better" by the novel's end.
Contributed by FUS RO DAH
BioWare produced six other covers for Mass Effect which were not used as the final cover of the game for various reasons.
Contributed by RayMonkay
On several explorable locations in the game, namely Altahe, Nepmos, Nodacrux, Rayingri, Eletania, Xawin and even Luna (Earth's moon,) strange sounds can be heard when the player exits the Mako and stands in a specific point of the map.

Although never confirmed by BioWare, the Mass Effect community has speculated that these peculiar noises are 'songs' from the insectoid and believed extinct race the Rachni.
The planet Amaranthine found in the Fortuna system of the Horse Head Nebula shares the same name as the city in Dragon Age Origins: Awakening. Both games were made by Bioware.
Contributed by FUS RO DAH
The Shifty Looking Cow is a Space Cow found on the planet Ontarom, in the Newton system of the Kepler Verge. An otherwise normal member of the species, the Shifty Looking Cow likewise has a dialogue option that does nothing. When Shepard turns around, however, the Cow promptly begins stealing Shepard's credits slowly, and will even follow the Commander around. It can be found near the southwestern corner of the map, near a deposit of gold.
Contributed by FUS RO DAH
On the planet Noveria, when the player attempts to restore power to the Peak 15 facility, the virtual intelligence "Mira" immediately appears and says "It looks like you're trying to restore this facility. Would you like help?" This is a nod to Clippit, the infamous Microsoft Word paper clip helper character, who was infamous for popping up and offering unneeded and unnecessary help. Shephard's Renegade response is "Oh crap, a pop-up."
Contributed by DidYouKnowGaming