This is definitely one of the best quests in the entire game and I loved every minute of it. Read on to find out how to start this quest. I will not post any spoilers in this post and will not allow any comments that might spoil the quest.
How to start the Silver Shroud quest
You must listen to the Silver Shroud radio near Goodneighbour. Once you do this then quest will appear asking you to go to the Memory Den to speak to Kent Connolly. Kent will ask you to collect a costume from the store “Hubris Comics”.
I have already collected the costume but cannot start the quest
If you collected the costume before starting the quest/listening to the radio then you must go and speak to Kent Connolly manually to start the quest. You can find Kent Connolly inside the Memory Den. Once you enter then take the door on your right and he will be next to a radio.
I hope this helped, I had already collected the costume before starting this quest which made it a lot more annoying to find. The quest is worth it though, it is one of the few quests in Fallout 4 with some interesting choices.