Shyvana is a tanky dps that is now one of the strongest initiators in the game, she has the potential to get a lot of early game kills.
DO NOT think this build is going to carry your team out of elo hell.
DO NOT rage against this build because you think "it sucks"
give constructive criticism if it didn't work for you
question the build, feel free to ask my why I did this and that if I have not explained it well enough.
And yes I know she is a good jungler but I think she is even better in lane

I have played Shyvana in about 30 games stats:
First blood: 10 times
Times jungled: 1
wins/loss: 29/1
Times had more deaths than kills: 0
ranked games: 3

Reds: I like to begin with physical damage for two reasons
1: I love the extra damage it gives you when combined with everything else. You end up with around 90 damage at level 1
2: vs armor pen take damage mostly because only one of her abilites actually does physical damage and that is her q which is just two melee attacks, her burnout spell uses attack damage ratios but does magic damage.

Yellows: You can go either armor or more damage depending on whether or not you feel safe with your team, if you think your team is good enough to support you then go damage. Another perk of having armor is that you can clear minion waves very quickly at very low levels because the minions do hardly any damage to you.

Blues: Magic resistance per level is good because this build doesn't have any magic resistance items so it ballances the atma's.

Quints: You could take really anything but more damage is always good.

These masteries seem a bit odd but they work because you have survivablity and damage at the same time. extra health, extra damage, armor, mr, armor pen, cdr, all very nice.
Not really much to say here.

Ok the biggest part of any champion is the items so here we go

Start with a doran's blade because if you stick with this whole build then you will start with 90+ damage and around 620+ health. I got first blood on a ryze in 4 hits in a solo lane vs him and a tryndamere by just walking into the bush and autoattacking with burnout on, didn't even have to go back for health.

Next is boots like any good champion, you can wait as long as you like to get these.

Phage: This is fantastic because 1. it gives you some much needed cc 2. it stacks more health (WOO MOAR HP) 3. MOAR DAMAGE

Ionian boots: I can already hear you all going AHHHH WHY NO GET BERSERK??? I DON'T EVEN??? The reason you don't get berserkers on her is because she is not a melee hitter, in that she doesn't just sit there and hit her target like tryndamere or riven, she bounces around her target jabbing them and letting her abilities do all the damage, the faster your burnout comes back up the faster you have more CONSTANT damage.

Sheen: just the next item towards trinity force and it has a good proc, that's all.

Ok here's where you can start to branch off depending on how the game is going. If you're destroying (like you should be) then skip the brutalizer and go tirnity>cleaver>atmogs (atmogs meaning atma's impaler and warmog's armor combo), if you are going alright but not great then build Brutalizer>ghostblade>trinity>atmogs>cleaver. If you are doing poorly or your team is just feeding then I'm sorry but I don't know what to say because I can honestly say I haven't been in that situation before. I will update if it happens.

Take burnout at level one and spam it down to your lane so you can get into the bushes faster. It also does a lot more damage, overall, to enemy champions and minions.

Twin bite at level two, max burnout asap, when you can't level burnout put one point into flame breath. all others to twin bite.

Flash: great spell for catching enemy champions and running away. Seriously that's all it's good for

Ignite: It's ok if you really want to secure a kill on someone running away but other than that it is kind of useless unless the entire enemy team is mundo and fiddle.

Teleport: not really needed since burnout gets you back to your lane so quickly anyway.

Cleanse: good for getting out of sticky situations and, come season 2, will clear other summoner spells like ignite.

Exhaust: good if you need to catch a single person but think about this: You're running away from the entire enemy team because you were caught (silly you) you flash over the wall and even when they flash after you all they see is a trail of fire leading off into the distance. Now if you have exhaust then you could slow one person and four of the will jump you and you will still die. On the other hand if you are chasing the enemy team WITH your team then you can slow one person enough for your team to catch up, securing the kill. Really your choice

******* it's easy to farm with her, seeing as she doesn't have mana you can spam everything without fear of becoming useless. Simply turn on burnout and hit the minions to extend the duration. If you have even a basic knowledge of when to last hit a minion then you will be fine.

W.I.P build, I may come back and edit things depending on patches and fixes etc.