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.

A few additions to Jeff's advice if you decide to replace the damaged drywall.

1. I'd suggest you look into waterproof drywall or mold resistant drywall. There are various products that do a better job than the typical green board that you would use in the bathroom that will help if you are installing in an area that has a lot of exposure to the elements.

2. Double check the thickness of your drywall if you are patching. If you replace a 5/8" with 1/2", you'll notice it.

3. Use screws, especially on the ceiling. They take more time, but hold better and are less likely to cause the nail pops that you are seeing.

That being said, I think it's entirely possible that you're just more aware of drywall issues and there may not be anything wrong. To know for sure, I'd go up to the attic, pull up the insulation where you suspect damage, and look for water rings or other mold growth. Use a flashlight, no need to feel with your hands (and risk falling through). While you're there, check the insulation for blackening that would indicate mold is growing there, and if so, replace any damaged pieces.

Your ability to see the joints between the drywall is just a sign of not enough paint and/or primer. And popped nail heads are normal as a home moves. My personal plan is to replace all the popped nails with screws and double up on the spacing the next time I paint to minimize this issue.