AP shyvana is fun to play. People like to point out that only 2 of her abilities scale off AP. Yes that's true but people forget that 3 of her abilities deal only magic damage. Which is why stacking magic pent items is the best build for AP shyvana, rather than just stacking AP. Still, your E will hit like a truck and you will burn through the other team with your W. Double actives are so much fun nuking power. And The new deathfire grasp is amazing now that you dont have to waste gold on mana regen and gives some nice CDR.
I've included 3 guides, in order of increasing trollfulness: jungler, solo top, and ap mid
AP jungle Shy
In general just follow the item summary. A few options depending on team comps and how the game is going:
1. You can just sell your cloth armor instead of building into the Emblem -> Locket
2. The core of this build is the Deathfire grasp, rageblade, and crystal sceptor. Everything else is just whatever you want. Besides the game will probably be done after you get those 3 items.
You need to smart cast your active spells. Use them along with your Q for INSANE BURST, then melt their face off with your burnout. That sceptor is what makes AP shyvana OP, it is a perma-AOE-slow with your W. Ult into a team-fight with your W on and you just slowed their entire team.
AP On-hit Top Lane
Malady is a pretty good item no one uses so I figured I should add it cause Shyvana works great with attack speed items. If the game goes really late sell the Haunting guise for sunfire cape. It recently got buffed to deal 40 magic damage a sec and works great with burnout. Wit's End is kinda of a standard item for Shyvana on most builds so I added here for more item-generated magic damage.
AP mid shy (warning: slightly trollish)
This one is kinda of just for fun. The objective is to stack so much magic penetration that your burnout (W) will melt the entire opposing team. Along with your Sceptor you will be slowing the entire team. Use your two actives, Q , and E to burst the squishies.
You will be doing so much damage that they can't ignore you. Unlike off-tank shy where they kill your AD carry and forgetting you cause you lack high-damage and strong cc, in this build they either focus you or get megaslowed and destroyed. If they focus you just have your Vayne clean shop. ENJOY. Try it out before voting. Thanks!
ps if I havent said it enough already, the AOE slow of burnout with rylia's crystal sceptor alone is worth trying this build out. Its kinda of OP. Play it on normal though, on ranked you may get reported for trolling :)