The simplest and most obvious multiplayer mode would be quick-run:
Players spawn at the beginning as usual, and compete with each other to finish the level as fast as possible.
Players cannot interact with each other, cannot steal powerups or kill monsters. You are just seeing a 'ghost' of other player.
This mode allows for more than two players, and is not influenced by network lag at all, it might be interesting to new players, but I feel it will grow boring quickly.
Deathmatch/CTF: will require specially crafted maps, very sensitive to network lag, the gameplay is not designed for deathmatch at all, and will essentially be a glorified
Divekick.
This is probably the worst choice of online gameplay.
Defense: half of players are playing as Tux, another half are playing as various badguys scattered across the level and trying to stop Tux from finishing the level.
Badguys respawn after being killed, Tux players stay dead.
So if you received a snowball, you can suicide and respawn as a jumping Spiky.
All badguys can jump, but Spiky will move more slowly, and cannot stop jumping.
Ice cubes require other players to launch them sliding, but can stop on their own after two seconds of sliding. Alternatively: ice cubes can slide but cannot jump.
When all Tux players are either dead or finished the level, the teams swap roles.
For quicker matches, only one Tux has to reach level end to win the match, then teams compare their run times.
Influenced by the network lag the same way as DM, but it will feel less 'unfair', because the teams do not have equal roles.
Players on the same team cannot bump into each other, because doing precise bump-and-jump physics will require very small network lag.
Players also cannot steal bonuses for the same reason, meaning that if one player takes the bonus, other players still can take the same bonus on their screen.
Infection: The same rules as Defense, but there's only one badguy player initially, and every Tux player who dies becomes a badguy.
Tux players who reached the end of the level respawn at the beginning, maybe with a bonus snowball.
Player who completed the level most of the times as Tux receives the best score.
If you'll implement online game mode, I will arrange two servers for it, but I will need SuperTux to be able to run in dedicated server mode, with no GUI.