What are some of the side-quests related to your background?
I've already discovered
Sole-Survivor Colonist: Suicidal Woman on docks
War Hero- Earthborn: Former gang member appear for your assistance
What other background sensitive side quests are there?
There is also the Spacer quest, where you find a washed-up bum with PTSD that served under your mother's command. There are also some normal quests you can do with any service background, but certain quests will give a more in-depth look into the events Shepard went through, and people involved with it. I know for War-Hero you are lured into a trap by the leader of the Skylian blitz, which you fought off almost single-handedly. For sole-survivor you confront a corporal that you thought died under your command along with the rest of your unit at Akuze. I'm currently working on an Earthborn/Renegade playthough, so I don't know what it will be this time.