How do I find the source of the water leaking through the ceiling?
All the advice so far about ice damns is valid.  However, if there are no dams or visible water on the roof, then you will need to crawl up in the attic and start looking for the wet areas that are wicking into your ceiling.  The insulation may have been soaked before you got a chance to clear the roof. Obviously if wet insulation is found, it must be removed.  Another problem I have come across in a few situations like yours, no dam, but slow moisture stains keep developing, came from improperly installed "proper vent" from the soffit vents to the air space in the attic.  In a few cases, the open upper end of the proper vent was blocked or covered with insulation and condensation formed on the top side of the drywall right along the wall. In a couple of cases, no venting was used from the soffit and insulation was packed into this area instead.  In either case, you will have to get up there and move the insulation in the affected areas and look for the moisture.  Also look for any tell-tale water stains on the rafters or trusses that may have been a leak from above.