I'm not going to talk about what happened to Shepard in my headcanon, since that story is told in Invictus, the link in my sig. For the record, I didn't pick Destroy and I didn't pick Synthesis - I'm sure that you can fill in the gap 
Ashley (Romanced): When the Crucible fired, it disrupted the Mass Relay network, stranding the Normandy in uncharted space. During the long voyage back to Earth, Ashley discovered that she was pregnant. Within a year, she gave birth to a girl, naming her "Cassandra". She resigned from active service in order to raise her daughter, and took up a teaching position on Elysium. The Williams family still keeps in touch, and Ash is regularly seen at Earth, the Citadel, and at the rebuilt Arcturus-Memorial station.
Garrus and Tali: Interspecies relationships are uncommon, but not unheard of, and Garrus and Tali got married two years after the Reaper war. As the quarians settled onto Rannoch, the Admiralty Board was replaced by a planetary government, and Tali'Zorah took up the position of Council Ambassador. Rannoch is outside of Council space, and neither the quarians nor the geth have any particular desire to join the Council, but diplomatic relationship is strong, particularly with both humanity and the turians. Garrus was offered a position as a Spectre by Councillor Sparatus, but after some consideration turned the offer down. He regularly arranges reunions for the Normandy crew, often holding them on either the Normandy itself, or at the Citadel flat that he inherited from Shepard.
EDI and Joker: Human-AI relationships are unheard of, however, but that didn't stop Joker and EDI. The two of them still work together, with the SR2 having been repaired and refitted by the Alliance military after the War. Questions about their private life are met with a joke by EDI, or by evasive comments from Joker. AIs are still illegal under Council law, and the Council was not pleased to discover EDI's existence once the Reaper War had been concluded. Ultimately, however, they ruled that she had a right to existence - the law hasn't changed, but she is now an acknowledged exception.
Liara: Liara dismantled the Broker's network after the war, and turned her attention back to archaeology - putting new research into the revelations from the Reaper War, and writing "Journey's with the Prothean" with Javik's assistance. She ultimately accepted a professorship at the University of Serrice. Privately however, Liara kept some of Broker's contacts active, using them to keep a watchful eye over the galaxy as it rebuilds, and to look into the Reapers' mysterious disappearance.
Miranda: Ex-Cerberus, but also a known associate of Commander Shepard, Miranda was a controversial figure in the Reaper War's immediate aftermath. The Alliance originally intended to prosecute her, but at trial it came to light how her actions had shut down Sanctuary and allowed the successful assault on Kronos, changing the course of the War. She was given a full pardon, and, in co-operation with her sister Oriana, Miranda began a new tech-company called "Lawson Consolidated". The company research has recently begun looking into practical applications of nano-technology, with Miranda utilising her experience from running the Lazarus Project.
Jack: Jack is still teaching biotics at Grissom Academy, and is surprisingly one of the most popular teachers. Some of her older students tell stories of fighting alongside her in the war, impressing freshers at the academy, although these tales are often exaggerated... probably.
Kahlee Sanders: Kahlee Sanders, meanwhile, never returned to the Academy, disappearing from the public eye for a short while before running for the position of humanity's new Councillor. She succeeded, and has helped humanity become a constructive force in rebuilding the galaxy.
The Krogan: Contrary to Salarian fears, the cure of the genophage did not lead to a new set of Krogan rebellions. Under the combined leadership of Wrex and Eve, the Krogan have reclaimed a number of their old colonies, but have surprised the galaxy by showing an impressive capacity for artwork and song. The Tuchunkan pyramids, recently completed, have become a popular tourist destination, built as both a memorial to the Reaper War, as a symbol of hope for the future.
The Geth: The Consensus welcomed the quarians back onto Rannoch, and that alliance has remained strong. The galaxy at large is still wary of the Geth, but they have successfully opened trade routes with the Alliance and with the Salarians. The Geth have also started a research corporation, naming it "Corpus Consensus", looking into transhumanist technologies such as synthetic limbs and brain implants. They often find themselves in competition with "Lawson Consolidated", but the two companies have worked together on some projects.
Zaeed: Zaeed kept trying to retire from work, but somehow it never really stuck. Despite now being in his mid-70s, he is still a mercenary force to be reckoned with, although most fellow mercs will refuse to take job with him. It's not his attitude that they have a problem with - they're just afraid that they might not make it out alive...
Kasumi: Kasumi became mysteriously rich after the Reaper War, at around the same time that various human companies previously associated with Cerberus collapsed. She now owns a home on Terra Nova, with an extensive library and a personally designed garden. Her neighbours say that she seems to come and go at strange times, however...
Steven Hackett: Widely praised as a war hero, Hackett took early retirement after the Reaper War and moved to Earth. He lives a fairly private life in New Zealand, but is known to have developed a reputation as a Sky-Car collector.