I'm going from serious to completely bonkers.
The political situation on Earth is still a clusterfuck, despite the discovery of the Mass Relay and confirmation of the existence of inteligent alien life. It takes the Reaper war to unite the whole world for good.
After the Crucible is activated the Quarian ships provide the large majority of the food supply to the Turian fleet trapped in the Sol system. This ends up being a great political development for the Quarians, who become an officially recognized Council space species with an appointed Councilor once the first relays get back on line.
Shepard is profoundly depressed throughout the whole trilogy and suffers from PTSD because of the background event, whatever it is. That's why the Commander is so understading of other people's personal issues and deviates from important missions in order to give his friends some closure. The willingness to enter any life-threatening situation is born of a troubling desire to end his/her life.
Eve is not actually attracted to males, but will nonetheless agree to mother as many children as possible for the future of her species.
The reason Legion used Shepard's N7 armor to repair itself was purely a design choice. Legion liked the stripes and logo, and thought they made the platform look very distinct and cool.
Admiral Hackett is Hannah Shepard's baby daddy if Shepard is a Spacer (we talk to mom but dad is never mentioned).
If Shepard is male and romances Liara, he's her first.
Miranda giggled throughout the whole portion of the Lazarus Project that involved recovering Shepard's dirty bits. She kept recallling the line in the Illusive Man's orders that had her repair all organs and functionality of the Commander's body.
Mordin's essence fertilized more eggs than he could ever count. He never bothered with reproduction negotiations but he provided the Salarian government with his exceptional genetic material to enrich the Salarian galactic genepool.
Wrex has a bad back and suffers from chronic pain but he keeps it quiet to save face. He secretly smokes an Asari equivalent of weed illegally acquired through Harrot's extranet discrete mailing service.
Every morning Garrus practices drawing his gun in front of a mirror, in his underwear and socks. Still looks adorable.
Kaidan waits until everyone is done before he showers because he's embarrassed about his size. He's actually big enough to cause embarrassment to others and got fed up with people's gossip.
Ash feels guilty about being attracted to Garrus.
At first.
While serving in the SR1 she starts to write poetry about a mysterious former Turian law-enforcement agent turned vigilante who takes matters into his own hands when the law doesn't know how to deal with the worst scum of the galaxy. She keeps it locked and safe with several passwords. This is before Garrus becomes Archangel.
Tali's handwriting is terrible and no one can read it, not even her.
Zaeed loves pussycats, sort of like a space-Hellboy.
Kasumi has some serious Freestyle Rap skills.
Samara is the best in the sack but no one will ever know because of her oath.
Jacob is not actually that boring, he's just playing that part because he was ordered to by the secret agency he works for, and whose prime interest is to discover the true location of the Illusive Man and mess with his weird tech-eyes' software drivers, so TIM won't ever notice Jacob drew a glowing dick on his forehead. The purpose is to discredit TIM through humiliation and make it impossible for anyone to take him seriously in the future.
Thane asked his son to take his body back to his home planet and spread his ashes in the same desert as he had spread his wife's.
Joker has a huge boner for Traynor but knows he will never be her type. He dreams of threesomes with her and EDI.
Grunt cried out of happiness when he had his first donut.
Vega and Cortez have a very hot night of don't-want-to-be-alone-tonight-sex after the Crucible was fired, but Cortez decided not to tell Vega it wasn't that good. They never had sex again either.
After the war Jack opens an animal sanctuary and actively seeks and destroys people who torture them for fun.
Kal'Reegar wanted to propose to Tali after they met on Haestrom.
Javik refuses to bathe. He doesn't wash anything more than his hands. The crew is too wary of him to say anything.
Dr Chakwas had some racy pictures taken when she was a young med student, that some people threatned to reveal when they learned she was serving in the SR2. Since she had no family to speak of she told them to go ahead and publish them, since she was quite the bombshell as a young woman and was anything but embarrassed about her body. Zaeed bought the negatives immediately.