For this list, I've taken the Fallout 2 set of skills, because I think that is the most complete and versatile set within the series. I've also seperated the combat skills from the non-combat skills because of their likely similarity to how they'd run in the single player fallout games.
[BIG GUNS]
Would likely run identical to other fallouts.
[ENERGY WEAPONS]
Would likely run identical to other fallouts.
[MELEE WEAPONS]
Would likely run identical to other fallouts.
[SMALL GUNS]
Would likely run identical to other fallouts.
[THROWING]
Would likely run identical to other fallouts.
[UNARMED]
Would likely run identical to other fallouts.
[DOCTOR]
This skill would most probably allow you to heal your physical injuries, like crippled limbs, etc. It might also allow you to do the same to NPCs and other players (which would make grouping more interesting, having a skilled medic in the group).
[FIRST AID]
Probably the same as Doctor, though administering and using chems, instead of healing phyiscal injuries.
[LOCKPICK]
I'm not sure what additions could be made to this for multiplayer. There would obviously be a call for lockpickers out in the wasteland, but what could you add to make it more multiplayer friendly?
[REPAIR]
Pretty much the same as single player, though perhaps you could fix up vehicles, and use them to travel across the vast wasteland quicker?
[SCIENCE]
You would be able to crack passwords and reprogram computers, but who knows what else it could be used for? Perhaps you could mix and make your own chems, and sell them to players. Addiction could mean a constant need of these chems - profit for unscrupulous scientists.
[SNEAK]
Obviously, being able to move more undetected around enemies and players (perhaps it could quiet your footsteps around players, making them less likely to hear you?)
[STEAL]
Perhaps this could involve a UO style thieving system where you could steal items out of the inventories of other players? This would have to be very, very, very well balanced; or else every town would just become a den of thieves. Perhaps different towns could have different outlooks on stick fingered citizens, leading some towns to become safer than others? It would be brilliantly immersive if you avoided the seedier towns and locations; because it actually meant you might be robbed or murdered if you went into them.
[TRAPS]
Setting traps for players, enemies and creatures? This could add a new layer of strategy to combat.
[BARTER]
I'm not sure whether this could be adapted to trade with other players...but certainly it could influence trade with NPCs.
[GAMBLING]
If there were casinos in game? Or perhaps you could gamble with other players, anywhere out in the wasteland. It would mean there was something to do while you waited around.
[SPEECH]
Of course, Dialogue options with NPCs would be handy, but this might have effects on players too.
[OUTDOORSMAN]
It would be excellent if you could make a living as a hunter or a trapper, hunting Geckos etc, selling the pelts to Tribals, or for crafters to make things out of?
I've only suggested a few, so, anyone got any additions?