As a general rule, each Green Lantern has one green ring.
This is more than just a general rule for the Green Lantern Corps, in fact: with the exception of the Orange Lanterns (all of whom are Ring Constructs of people Larfleeze has killed and/or eaten) each Lantern is a member of a single Corps and has a single ring.
There are very few exceptions:
Mongol briefly possessed several Yellow Rings, which he though made him stronger (and given your belief partially powers any ring, may have been true) before he was beaten into the pavement by Sinestro.
Guy Gardner was once possessed by a Red Ring. It didn't last long, Saint Walker (I believe) helped him purge the red energy from his system.
The Star Sapphires had a habit of kidnapping other ring bearers (especially those with yellow rings) and 'Converting' them. They were encased in sapphire and essentially brainwashed to serve as Star Sapphires. A few of the Sinestro Corps (yellow lanterns) converted to Star Sapphires (violet lanterns) this way, though they then stopped wearing their Yellow rings.
Hal Jordan briefly had a Blue ring forced upon him. It didn't let go until he had mastered it (by telling it what he Hoped for) and then it was released. Hal didn't want the blue ring to start with, and wearing two colored rings interfered with his ability to use either properly.
Hal has also worn a red ring (briefly), it seems:
Frankly, I'd be willing to guess that Hal has worn more rings than anyone else, because the War of Light was seen by many writers as a way to dress Hal up in as many costumes as they could.
All of the above happened during the War of Light. Further back in time, Sinestro gave up his Green Ring in favor of the first Yellow ring (alternatively, as the backstory changes, he was given the boot from the GL Corps and forged the yellow ring).
Guy Gardner wore a Yellow ring for a while, back before he became the Warrior (let's not talk about that, ok? It was the 90s, it was a bad time). This was the ring he used when he went up against (and was put down by) Doomsday. He couldn't use it very well (he didn't speak the language it used) but he made it work well enough. Batman still put him down with one punch.
Long story short: it isn't common for any Ring-based hero to use more than a single ring at a time, or for them to use rings of different colors simultaneously.
During Blackest Night, Black Rings found dead (and later, formerly dead) characters. Many dead Lanterns were risen as Black Lanterns, but they didn't retain any of their former colors.