Alright so I've been playing and testing different builds with Shyvana since she was released and so far I've found this to be the most effective way of achieving a decent hybrid build with high damage output. It is lacking in some defensive areas such as hp and resists but you can easily substitute for the game you're in.

Pros:
High Damage Output
Moves Quickly
Fairly decent hybrid
Can solo jungle early game
Solo dragon and baron!
Good farmer
She's a dragon!?

Cons:
Squishy
Has to be right in the middle of the action to perform well

Both of these segments are not in stone at all.

I chose these runes simply for their direct effects, really no explanation needed. You may want to substitute for some magic resistance/HP/More AS but that's up to you.

Not much to say about the masteries, you may want to take more cooldowns but I don't find it necessary or you may skip the dodge for some of the deeper defense tree items

Now here is where you can have some fun. I've tried a heavier AP build but it doesn't scale well so I found this to be a nice medium. Some suggestions for some of my items would be items with Attack Speed, Damage, Hybrid, Resistance.

I open with boots and health pots so she can get away and stay in lane longer.

Then I get Berserkers or Mercury Treads depending on how AP heavy the team is. I take berserkers if there's only 1-3 AP, or none in my lane.

Then I start getting synergies for Madred's Bloodrazor because this is going to be her main damage item. This really makes her basic attacks and Q chunk health away, also good for jungling. A substitute would be Wriggles Lantern here.

Afterwards I start building Hextech for the life steal, damage and AP. Substitute here could be Wits end for magic resistance and attack speed.

Next is Trinity Force. This gives her some HP and bonuses in all areas so this is a MUST HAVE ITEM no substitutes for this one.

Finally Infinity Edge is great to really scale your damage through the roof and absolutely decimate. Pretty much required for any champion who has high attack speeds and damage. But you may want Frozen Mallet for more HP and slows.

To round her out I get sunfire cape for the AoE and Health. This can also be situated to the game or taken earlier.

Shyvana has Dragon Rage so when you select your ult you'll notice the red bar. When this bar is full she is able to transform and use this 100 "rage" to maintain dragon form. Before large team fights I recommend jungling or clearing minion waves to build this rage. Each attack adds to her rage quicker than letting it accumulate on it's own.

Pretty basic, take burnout first to optimize farming and escapes/chasing. Max this first and Twin bite second. Last should be your Dragon Breath as this is just to harass or finish off enemies running away.

I like to always keep flash because you never know when you may need to chase or get away over a wall or just in lane.

Then ignite just to get that runner to drop his last bit of health.

With Shyvana you want to play her with a tankier champion or someone who can save you if you're running away. I like to play the bushes and come in and out and take a few shots with Burnout on then run back. Let your partner do the bigger harassing and you finish them off. Play conservative if your laning opponents are ranged or more aggressive. It's better to let them chase you under your tower then unleash your damage than to chase them into the open and die in a few hits.

Shyvana works well with a tank. You should never play her as the main initiator just the dragon who comes in with the AoE and takes down those tougher champions in team fights. Her Ult can also separate champions from each other and singling out the mages you want to kill first ect ect...

Your best bet is to get in the fight and start with your burnout then twin bite then transform into dragon to finish up who ever is left. Be careful to watch your health bar because she is fairly squishy and easily targeted.

This was a quickly made build and I am open to improve/take suggestions. From what i've seen I absolutely love this champion and she is very capable of taking out multiple champions if played right.

Leave comments/suggestions and thumbs up!