Brown or rusty water is usually corrosion — either aging galvanized steel supply pipes rusting from the inside, or sediment inside your water heater's tank getting stirred up and pushed out through the hot water line. It shows up more noticeably after nearby water line work (yours or a neighbor's), after the water's been off and back on, or right after running hot water following a period the heater wasn't used. If it's isolated to hot water only, the water heater tank is the likely source; if it's in both hot and cold, it points more toward the supply piping itself.
Cloudy or milky-white water is usually the least concerning of the three: it's typically just trapped air in the line, often after any work on the plumbing system or a pressure change. The simple test is to fill a clear glass and let it sit for a minute or two — if the cloudiness clears from the bottom up, that's air, and it's harmless. If it stays cloudy or has a different texture, that points to something else worth checking.
Blue-green staining, especially around fixtures or in water that's sat in copper pipe for a while (like first thing in the morning), points to copper pipe corrosion, usually from water on the more acidic end of the range reacting with the copper over time. It's worth having your water tested if you see this consistently, since it can also mean elevated copper levels in your drinking water. One more distinction worth making: if discoloration is affecting your water only, it's likely your home's plumbing. If neighbors on your street are seeing the same thing at the same time, that points to the municipal supply rather than your pipes — it's worth calling the city first in that case, since no amount of work on your own plumbing fixes a supply-side issue.