Everything I see mentions P-traps. I seem to have instead an S-trap. What are the downsides of this, and is it something that should be fixed?
You have an "S" trap because the drain exits through the floor and is probably unvented.
A "P" trap is for a drain in the wall that is teed for a vent stack.
The major disadvantage comes because "S" traps are usually used in locations where they are either not vented or poorly vented on the outflow side and so can siphon off the water in the trap, leading to sewer gas backing up into the building.
For a short term fix, after using any quantity of water, pulling the plug in the sink will empty it vigorously and pull enough water out of the trap to break the water seal, run a pint out of the tap afterwards to make sure there's a water seal.
For long term repair, you would have to install a tee on the drain, run the output from the P trap into one leg and a vent to who knows where from the other side. Plumber consultation recommended before you end up with one of those plumber's nightmare constructions.
It is illegal in most localities to have S traps in new construction because of the siphoning.