I know if the user is killed the ring seeks out a new host, but what if the ring itself is destroyed?
Breaking a power ring is like breaking a regular gun or a sword: the only thing that is broken is the ring itself (and as said by the others, they can be reforged). The main side effect of this breakage is that usually the wearer gets in trouble: unarmed and with a really limited defensive capability, he can become easy prey of whoever broke the ring. Also, as seen in the Injustice comics
being in space when the power ring was stolen (and broken) by Sinestro did not end well for Kyle Rayner
This can be seen in some comics: in the War of the Sinestro Corps power rings are broken and later reforged.
Another thing to note is that rings can be broken from overexertion (or at least in the New 52's GL series Sinestro says he has broken a ring once).
EDIT: You asked about all colors, so there are a few extra side-effects that I did not mention:
- As their heart is useless, Red Lanterns rely on the ring to keep them alive, so if you destroy the ring of a RL he will likely die.
- The Indigo Tribe is mostly comprised of psychopaths only held back by the ring's function to allow them to feel compassion. Thus, if the ring is broken they revert to their usual selves (as seen when the Indigo central battery is destroyed in the New 52 GL -vol. 1 if I recall correctly-)
New rings can be forged.
Each ring had to be created initially, and replacements can be made as well.
Rings are a potent tool, but they are not indestructible. When they are destroyed, the Lantern using them ceases to be able to use them (and frequently dies - not because the ring is destroyed, but because they're facing a foe that will kill them).
The rings contain immense emotional light power, stored within them. It is refilled regularly when the Lantern charges with a power battery (which can ALSO be destroyed). This power, however, is something that is naturally occurring in the universe, and doesn't interact with the universe outside of when it is shaped and focused by a Lantern (using their ring).
Rings don't explode or otherwise react violently to their own destruction. Most likely, the stored emotional energy simply dissipates back into the background emotional energy field of the universe.
The short answer is: nothing happens.
I can't recall any time in the comics this happens, but drawing from the movie "Green Lantern First Flight", which focuses mostly on the fall of Sinestro there is a scene where:
(after getting the yellow ring) Sinestro assaults Oa. Sinestro overpowers the Green Lanterns on Oa and forces them to surrender their rings. He then uses his ring to break each of theirs. As he does this the rings snap and shatter, but nothing more happens to them.
Back in v1 and v2 Green Lantern with Hal Jordan, the GL power ring was indestructible, as was the power battery. It was rare that a battery, or a ring, exploded from outside forces. With Hal Jordan becoming Parallax, all the rings and batteries "went dead", as they were supposed to be connected through time and space to the Central Power Battery on Oa. Kyle Rayner's single power battery was supposed to be "indestructible", as it was the last one.
When Hal Jordan came back in Rebirth and the GLC was reborn, the batteries got more easily broken, and with the retconning even Sinestro has admitted to breaking two or more GL rings in the past, from demanding too much of a power ring. After Blackest Night, the portable batteries are durable but often used as improvised bombs when the occasion demands it.
The indestructibility has been inconsistent throughout DC's history.