Support
You have by far the most damaging level 2, and as the game gets later and later, you can take on bigger and bigger threats. Late game you often end up trading your life for at least 75% hp off of one of their main carries and all of their carries' attention for 10-12 seconds. A Xerath/Kog'Maw combo is really scary unless they're both dealing with a Poppy, while your team wins the fights 4v3 and 4v"2". Of course, that's given the fact that you got few neutral objectives and mostly assists, otherwise we're talking about half their tea mdieing to you. I really feel like she's basically a playmaking Leona.
Jungle
Did I mention I like Poppy? Jungle Poppy is pretty simple. You have the burst for mid kills and 100% stun rates on side lanes. You basically rely on being Poppy and dishing out insane damage without ever getting coutnered by that awkward top lane udyr or a god damn olaf pick. If the enemy is pushing in, your ganks are easy, if they're not, you aren't exactly unhappy to farm.
Support
Farm that Q with targon, otherwise you simply deal hefty damage wit ha support thanks to those free APs and the Q and you have an insanely long AoE cage/stun. It's not uncommon to see Veigar come out of lane with 1k+ worth of APs on his Q (which is pretty cool when most supports have 1 completed item by that time).
And, last but not least, the one I'm currently trying out in Team Builder and getting extremely variable results (up to now, it'd be an excellent counter-pick).
Support
I'm feeling slightly lost itemization wise, but overall he seems to offer non-negligible poke as well as some ludicrous counter-engage. You are fairly squishy and don't have as much damage as a solo lane/jungle shaco, but it's all made up by your innate which adds 20% damage to a lot of your pokes/trades. JitB allows for great vision control as well as counter engage (Leona for instance cannot ever dash on you or she'll be feared and receiving damage from JitB and you have the time to deceive on her ADC before the stun). I've had some really mitigated results against Ashe and ranged poke-heavy supports (Zilean/Lulu mostly), but I don't think I've lost a single lane to melee supports.
Personally, I play him AD, basing my playstyle mostly around getting towers (you can take the tower solo in 1 recall if you have your ulti) and counter-engages, as well as picking up carries late game (kind of guerilla warfare). I will often itemize into an early Sheen into Trinity for that in-lane trading (we're ralking about 240% base AD thanks to Backstab, and the normal attack also is enhanced by both pseudo-crit and backstab). Then, I just buy Hydra+Shiv to make my clone even more powerful as well as give me an extremely potent split pushing power (skipping on a front liner/carry to deal with the support makes for some easy 4v4)
Oh and did I mention 1-man dragon aggro juggling? You can literally kill dragon at level 6 without receiving a single hit if you've got f**king Challenger's mechanics (I usually receive mitigate damages by roughly 20%, with my clone tanking up another 30-40%).