Side quests are those quests which are not faction-aligned in any way (e.g. given by Minutemen, Brotherhood of Steel, Railroad or Institute) or part of the main story. There are at least 18 of these, possibly more.
Some of these are provided by companions, some are triggered in cities or the wilderness. I've listed below all the quests I've found and how to trigger them.
*** Spoiler - click to reveal ***
Benign Intervention - visit the Combat Zone (near Diamond City), then take Cait as a companion and talk to her occasionally until she tells you about her drug addiction.
Diamond City Blues - enter Colonial Taphouse in Diamond City. You'll see a fight between two guys. Upon leaving the Taphouse or returning to Diamond City later, one of these guys will approach you, kicking off this quest.
Confidence Man - provided by Vadim in Dugout Inn in Diamond City.
Curtain Call - listen to the radio broadcast from Trinity Tower Radio (near Trinity Tower / Diamond City)
Detective Case Files x2 - chat with Ellie, Nick Valentine's assistant, once you've finished the main story missions involving Nick. There are two items in the office that will trigger quests.
Detective Case Files - a final mission with Nick, requires you to have completed the first two case files and have some friendship with Nick.
Vault 81 and Hole in the Wall - enter Vault 81 in the Western Commonwealth and gain entry (persuasion check or 3x fusion cores reqd).
Emergent Behaviour - Complete Vault 81 and Hole in the Wall, have Curie join you, and gain enough friendship with her to trigger this quest.
Last Voyage of the USS Constitution - from the robots on the USS Constitution in Boston Harbor, near Charleston (this is arguably the most amusing side quest in the game).
The Big Dig - given by Bobbi No-Nose in Goodneighbor, down an alley in the southeast behind two warehouses.
Human Error - Gain entry to Covenant (northern area of map), this should kick the quest off, or you may need to talk to Honest Dan within the town
The Devil's Due - find a holotape on a body near the Witchcraft Museum (Salem, NE corner of map).
Here There Be Monsters - talk to Donny, on the docks near Shamrock Taphouse and Harbourmaster Hotel
Kid in a Fridge - find the ... kid in a fridge south of University Point and east of Jamaica Plain (eastern edge of map).
Secret of Cabot House (x3) - this is a tricky one to start, but also essential to get the Charisma Bobblehead. Find Parsons State Insane Asylum (NNE of map) once you are at least level 10, and attack the guards. Now a ghoul named Edward Deegan will appear, but he could be either in Diamond City (Colonial Taphouse or Dugout Inn), Bunker Hill (the market) or Goodneighbor (The Third Rail). Talking to him will trigger this series of quests.
The Silver Shroud - Talk to Kent Connolly, in the Memory Den in Goodneighbor.