Hey this is my first character build so be easy :P
Shyvana isn't one of riots new OP characters this one actually takes a little skill and practice. she is fun i do enjoy her and she seems to be build more around attack speed over everything. I've looked at many builds people have been posting and this is what I have come up with

Pro's
The Pro's for Shyvana are :3

-Very deadly
-Fast mover
-Good Multi character
-Big time pusher
-Great end game
-Skill based character

Con's
The Con's For Shyvana are :(

-Focused
-No CC's
-Squishy early game
-Some long CD's
-Very Item Dependent
-Skill based character (said it twice but understand that she is not your face rolling character)

The Runes I have chosen seem to be the best set up that i have found out in some game testing

AP Marks: Well shes a Mele most Characters that do physical damage need the Ap for doing tons of damage and ripping through armor and it helps keep your items on topic for what you really need.

Attack Speed Seals/HP+ Level: Starting with alittle bit of attack speed doesnt hurt and it will give a boost just abit While the Hp+ level will keep your HP going as you do your self.

CD Glphys: Shyvana does not have any type of mana or energy so the faster you can click your flame wheel and your dragonbite the more damage you will deal in all :)

Ap / Armor Quints: I picked Ap because of Flame Breath / Burnout does Magic damage and with the AP boost i think there is real no harm or foul and can be changed if wanted. Armor Quinst because of most the items you are going to get will give some magic resist and not a lot of armor is involved so fighting AD characters will hurt a lot

The Masteries I went with are to be a AD Dps Shyvana with a few points swapped around she can be changed to an AP Shyvana but i don't think that would work to well and a tank Shyvana doesn't seem the BEST route to go in my opinion. She does have the potential to be a tank but I believe that she is one of those characters that are Suppose to be a Dps Class.

I took the Magic Pen Mastery because of Shyvana's Fire Breath / Burnout / Dragon's Decent. Those are her only Magic abilitys and i think that 15% Magic Pen should be enough not to focus it anymore than it is.

For my Summoners spells I went with Exhaust and Flash. but Shyvana can also go with a combo of ignite/flash.

Exhaust with the mastery to improve it is really nice when you want to burn down a single target in seconds with Shyvana and it is also nice for when your trying to escape and survive.

Ignite is also nice with the mastery point to increase your fire breath / Burnout's damage.

Then i went with flash because .... well i mean its nice when running away or just trying to get that kill or even for Shyvana trying to re-position yourself in a team fight to hit everyone while you are in your dragon form :3.

First I start with Burnout and I max it as fast as I can. This will be your main Aoe ability that will drop team fights down very quickly and even if your stunned with this up Champions will take damage if they are sitting inside of it. Not to mention in Dragon form it puts a path of flame on the ground in your path dealing additional damage so this will be a big part of your character.

Second I get Twin Bite and Max it after Burnout. This will be Shyvana's big single target damage dealing Tons of damage and not to mention your auto attacks will reduce the Cd on it so this is where your attack speed comes in handy. In dragon form when twin bite is use it hits everyone in front of her so in teams fights this will come greatly to knock down multiple People.

After those two are maxed out I finish Dragon's Breath. Dragon's Breath is a skill shot magic damage that reduces the Person's Armor by 15%. Its a nice skill but not one that needs to maxed out before your other two main ability's. In dragon form Shyavan's Dragon's Breath is just like Grave's Buck shot (Shoots out 3 fire breaths that pass through targets and reduces their armor by 15% also).

And lastly we have Dragon's Decent. Dragon's Decent is a dash / Jump that pushes Champions the in the direction she is going. It also Boosts her armor / magic resist by 15 / 20 /25 each level. It also Adds the extra factors to Shyvana's basic abilitys

I First start out with a Doran's Blade to get a little bit of life steal and hp going.

Then i start working for my Boots (depending on what you are fighting would determine your boots) if your fighting a lot of casters i do recommend you get the merc treds, if you are not fighting a lot of magic then i would recommend Beserker's or Ludicity (depending if you want more Attack speed or faster CD's).

Work for your wits end then your stark because Wit will give you a lot of mag resist and then stark will top you out with your Life steal and then get your phantom dancer to finish your attack speed.

I do recommend getting infinity edge last because if the round is going this long you'll need it and get the shield after that to replace your Doran's Blade.

In all together Shyvana can be very enjoyable to play or very frustrating. It all depends on how you play her and if you just like her as a character :).

She can be played many ways of course. She can be a very Good jungler, She can Solo her own lane, or she can just be that straight push in dominion or summoner's rift heck even Twisted Treeline if she is played correctly not to mention for Dominion and summoner's rift i believe this build can be used in there (with a few item changes that i will add)

I hope you all found this guide to be very helpful and would like some Feedback to see how I have done or how I may improve this guide some more :3