This common situation is most likely due to an actual drain odor. The rotten egg, sewer or sulfur smell, which is usually stronger in the morning and confined to a particular location, is a typical signs of drain odor coming from the u-shaped trap under the sink. As soon as the hot or cold water is turned on, the air-water interface in the trap is displaced and the odors are immediately released. The odor goes away when the water is run, but returns later.
To remedy this situation, carefully pour one cup of bleach down the affected drain and wait 15 to 20 minutes without using the water. Then, flush thoroughly with cold water. This treatment can be repeated, as needed.