Look inside the dormer for signs of leaks, water stains on ceilings or walls or around windows. Sketch an outline of the dormer on paper and mark suspicious spots, then climb on the roof with a ladder or through the dormer window and locate these areas on the dormer.
Check for loose or damaged shingles, with obvious tears, cuts or nail pull-outs. Look up the roof; water runs downhill and the leak may not be right at the dormer, but up the roof. Inspect seams, where two shingles meet; they should overlap and not be aligned. Lay a flat board on the roof to locate depressions where water may collect.
Inspect all intersections, where the dormer connects with the main roof. Lift shingles slightly with a pry bar if necessary to look at metal flashing for signs of rust or corrosion that will indicate leaks; remove shingles if necessary to inspect flashing. Look especially at corners; install special corner flashing that wraps around the corner to seal those junctions.
Test windows. Remove any loose or deteriorated caulk; take it off the entire width or length of the frame, even it is damaged in only one spot. Lay new caulk around window frames. Check around any wood trim, on corners. Replace any damaged trim and inspect under it for signs of deteriorated roofing paper or flashing. Caulk around new trim to seal it; don't forget the top, where the trim meets the shed roof line.