A rotten egg smell or sulfur-like smell might be coming from two sources or a combination of both:
Sometimes, when the water no longer has chlorine and other disinfection byproducts, the p-trap in sinks might develop some organic growth and cause the smell. The solution is to pour about 8 oz of laundry bleach or drain cleaner into the sink and let it sit for 15-20 minutes.
The rest of this article applies to tank-based water heaters only; it doesn't apply to tankless water heaters. The hot water smell does happen in some cases, depending on the make, model, and age.
The tank-based water heater's smelly water issue is caused by an "anode rod." It's designed to erode slowly before the hot water tank itself when the water supply is acidic or aggressive, thus extending the lifespan of the hot water tank. The municipal water supply is pretreated, and usually, the anode rod is no longer helpful. Certain life forms love the anode rod material and produce the smell as a by-product.
City/Municipal Water: The solution is also straightforward. Place the carbon and, if present, water softener tank on the by-pass (red/blue arrows facing/pointing at each other) for two days to allow chlorinated water in the hot water tank. After two days, take both water filter tanks off the bypass position (red/blue arrow in parallel.). Usually, the smell doesn't come back after one chlorinated city water treatment. If it does come back, then draining to flush out any possible debris from the hot water tank followed by another chlorinated city water treatment will help.
Well Water: In severe cases, usually on the well water, when chlorination of the plumbing doesn't help, the last resort remedy is to remove the anode rod altogether. On municipal water supply, it rarely comes down to it. This article does not describe the chlorination of the plumbing on well water.