If there is ice on the roof and it can be cleared do so.
Repair leaky roof in winter.
For example soak the downhill side of a chimney first then each side then the top on both sides.
The roof shingles should be softened and flattened as the shingles tend to curl up in winter.
The shingles can be warmed up with a propane torch.
For loose or damaged shingles remove them by pulling the nail out and scraping glue from underneath the shingles before replacing with new ones.
Electric tape works well but it must be applied before the winter season begins a pickaxe works well with risk to further damage to your roof.
Sealing the shingle to the object can be accomlished with a roofing sealant available at menards lowes etc.
The rubber gasket that surrounds the vent pipe could be rotted.
Roof leaks are confusing to diagnose.
Look for cracks in the base of flexible rubber flashing or broken seams in metal varieties of vent pipe flashing.
Using salt on the roof edge may be the safest and easiest way.
However when it is not practical to climb up on the roof to fix a leak such as during the winter months you can repair a leaky roof temporarily from the inside if you can determine where the leak is occuring from the underside of your roof.
Check plumbing vent boots and roof vents if water is entering from that area to cause the leaking roof.
Isolate areas when you run the hose.
Roof leaks must be repaired from the outside to ensure the problem is corrected.
For flat roofing look for cracks.
Start low soaking the area just above where the leak appears in the house.
Various conditions can cause roof leaks and the origination point may be difficult to detect.
To repair a leaking roof with curled shingles straighten the shingles with a heat source like a hair dryer and reattach them with asphalt roof cement or compound.
2plumbing vent boots and roof vents.
Locate the leaking source and search for the damaged or missing shingles and make a list of the areas to be repaired.
Make use of the flame carefully on the curled shingle edges.
Perhaps placing tube on furnace duct so air heats pre cut tube inside of home then climb up ladder and seal area keeping the sealant warm and easily caulk able.