Welcome!

This build is a work in progress. However, my idea behind it is to make Shyvana a Beefy DPS fighter.
This build is for aggressive players who like to run into a fight and be able to get out all the same- with a few extra kills then run into either your or your opponents jungles and still wreak havoc there too.
Success with this build requires knowledge of Shyvana's skills and when to use them, which means no dragon jumping into a 4v1 even when your fed. Learning why each skill is there and what order to combo with during an engagement.

Pros

Amazing DPS
Incredible pusher
AOE on every skill in Dragon form
Great Jungler and counter Jungler
Great chaser
Good initiator
Can be tanky
Shes.. A Dragon!

Cons

Focused
Has no CC
Cant sustain early game in lane

Obviously the Pros outeigh the Cons.. By the way did I mention shes a Dragon?!!

I went 21/0/9standard build because her offensive capabilities really need to be utilized. I put the rest into utility so that I could achieve the experience gain for early game advancement. Taking a hit will be a fundamental part of Shyvana, as well as attack speed and Q hits.

The 9 Greater Marks of Alacrity will further scale your attack speed especially earlier in the game when you put off getting youmuu's for the sustain of Phage.

The 9 Greater Seals of Resilience will give you more defense so that you can buy the boots first and guarantee the first gank that would've otherwise failed if they decided to fight or took you by surprise somehow.

The 9 Greater Glyphs of Focus will bring that cooldown of Twin Strike even lower than your passive is already doing so you can spam that between auto attacks, used with the boots of lucidity and youmuu's ghostblade you'll be fairly ahead of any other Shyvana.

The Greater Quintessences of Alacrity should be fairly obvious as to the choosing when DPS is the foundation of the build.

Flash

A truly important mastery for surviving and initiating ganks. Shyvana can use her ulti to initiate as well but there is that chance of failure and a flash to get out or finish off that last fleeing enemy is priceless

Ignite
I'll be honest, Ive become addicted to ignite and it's usefulness throughout the entire game, whether it be forcing an approaching enemy back, finishing a tower hugger/runner/flasher, there's just so many uses it's hard to choose another skill

Psychology is Important

Originally I thought that getting Flame breath to poke was the best way to go but chasing enemies with Burnout does so much more damage and speeds you up to the point that you will hit them for more damage than flame breath ever could.

Burnout is by far the most important skill to get!

After Burnout is maxed, Twin Strike will really be ready to do some damage with the Trinity Force, as well as be able to get spammed because of our attack speed.

Max Flame breath last because it's compounding damage is based on AP and the Hextech Gunblade is really the AP item to wait for to use this well.

Starting Out

Boots and health potions right out of the door will let you run in and last hit minions and get out of any skill shot's range or the range of click and hits better than starting with Dorans shield or Dorans blade. And why get a health crystal when you can just avoid getting hit instead?

Core

Getting life steal in the beginning lets those auto attacks give more and more. Shyvana lacks the health early game to sustain so getting some lifesteal over the phage puts her on a more offensive track.

Getting Battle Grieves over Mercury Treads is to increase attack speed early game because you wont see it jump until you build something out of the Recurve bow.

The Phage is made to start off the creation of your Trinity Force because the slow down combined with her eventual attack speed will work at all parts of the game.

Trinity Force was originally out because of the wssted money and buff slot of mana, but the Sheen's passive is too much to give up. And the finished Trinity Force just makes it better. The AP will help give abilities some damage as well because the abilities will become ineffective late game, especially flame breath.

Situational and End game

The Chain vest is there to let you decide whether you need guardian shield or Thorn Mail. I find that there are always more AD carries nowadays so I usually end up builing the thorn mail. However, if your the one destroying get guardian shield so you can get back up and destroy some more

Hextech gives you the full potential that the Vampiric scepter has been giving you all game. Not only that but the large AP boost will allow your abilities to catch up some with the overall attack power your dishing out.

By this time you should have won or are fairly close so why go for a really expensive item when you have so little time left to build it? Get that little push over the edge and get Recruve bow to just make you hit that much faster which then can be built into four different situational items.

Hit and Run

A very important note to consider when building as I do is to compare how you actively play. If your used to chunkier fighters or tanks you probably should turn away. AP - Get out. But if you prefer the initiating/there and gone approach then read on!

Laning Phase

During early game, assuming you top solo or with an ally, you should focus on spamming burnout and running towards the enemy, making them flee back a bit then turning on the minions again. Knowing your area control is important; staying near the bushes at the top is vital or assuming they control them with a Shaco or Ashe; you should remain in the lower corner- right behind your minions. This adds a buffer to most forms of skill shots and a semi-perm wall to slow enemy advancement.

6-Mid Game

It can be quite a laugh to be in the middle of a 1v1 at level 5 and slowly falling behind to light up and ascend to level 6 with full rage just in time to jump into Dragons Descent and crush that tyrns hopes and dreams.
Dragons Descent is amazing but it does take some time to rechage, so your gonna be wanting to be hitting something all the time. This means turning into a roamer. The scariest form of a champion. To live in the jungle and the lane and be able to solo gank. Hitting the golems and other monsters is important if your jungler is out and about too. keep hitting minions and monsters and enemies and the Dragon will never let you down.

A word of caution is that even though there WILL be carries, tanks, other DPS, AP nukers, Glass cannons, and Fed allies.. YOU will be focused lol. Even though it doesnt look good and can make you angry in the end your team comes out on top because you jump in the middle of their group with dragons descent, burnout and start spamming abilities your AOE will be most effective, plus your allies will be taking them out from their newly lowered health levels.

Late Game- Win
(Because Losing is not an option.)
-In Progress-

Outro
So Shyvana is pretty much an awesome DPS killer that we've made sustainable and hard-hitting. All we can do now is improve our use and timing with her skills and hope we don't fail her because her and her items aren't to blame!
She can be built differently if you prefer jungling or mid solo but I prefer the reassurance of a teammate and ganks so that I can be aggressive and do those annoying hit and runs everyone loves. Shyvana is a fun and rewarding hero so play well!

By the way did I mention Shyvana is a DRAGON?!