First off, this is my very first build I have posted on Mobafire. I use this in Ranked, 5v5, and 3v3 games. I have been playing Shyvana since the very first day she was released and have quite a lot of experience with her. This build focuses on giving you lots of armor and magic resistance while still taking advantage of the decent amount of damage Shyvana can do purely by auto-attacking.

My masteries are a relatively simple 21/9/0 build.
In offense I go all the way down from Fury to Executioner stopping along the way for Destruction and Havoc for the added damage output and ability to take down turrets quickly. I also like to pick up Butcher for early last hitting and late game wave clearing.
In defense I take 4 points in Durability and 1 point in both Resistance and Hardiness for survivability and early game tankyness. I also take Tough Skin and Bladed Armor to take the edge off of Jungle Monsters and minions.

These runes are somewhat standard for a tanky-dps champion. Greater Mark of Attack Speed and Greater Glyph of Attack Speed take advantage of Shyvana's great ability to auto-attack a champion to death through out the entire game. It also helps keep her Dragon's Descent active for a longer period of time and use Burnout to it's full advantage.
Greater Seal of Vitality makes you tanky early game and makes it really easy to bully enemy champions out of the lane. Not many champions are going to be able to keep up with the extra health early on and it is a nice buffer late game against effects like Fizz's Seastone Trident and Ignite.
Greater Quintessence of Vigor keeps her in lane longer through the whole game which is very useful early game. Especially when you are able to decimate entire minion waves while you wait for your recently dead teammate to get back in lane so you can recall.

Let's start with the summoner spells. On most champions I would take Ignite and Flash. But I have found that as Shyvana you are barely ever in need of Flash due to her lack of ranged moves. You can use Dragon's Descent to pursue or escape if you really need to and if you are just barely behind a champion you can always use Burnout.
My reason's behind running Exhaust and Ignite on Shyvana are relatively simple. With Ignite, you can pick up kills when an enemy champion is trying to make an escape after they have taken a beating, or to throw a wrench into the works when a champion like Volibear has it's passive go off and potentially win a battle or escape.
While I used to take Heal as my second summoner spell as Shyvana, all the recent nerfs to heal have made me rethink that decision. Now I almost religiously take Exhaust. This will help you out damage someone in a duel, stop someone like Vayne from facerolling you and your allies, and if the need should ever occur, you can even slow someone to catch them.

Twin Bite is an excellent move. I take it at level 2 and max it second behind Burnout. This move does loads of extra damage and can even be used on turrets. Now, it doesn't do tons of damage to turrets. But it gets them down just a little bit faster.
Burnout is your best move. Not only can you clear entire minion waves with it in seconds, but it also does extra damage in a fight, allows for escapes, and helps you catch up to enemy champions with low health possibly killing them by simply following them if they are squishy. Max it first and you will reap the rewards the whole game.
Flame Breath is Shyvana's least appealing move. It hard to land and it's duration is so short that it is hard to do enough damage to make it seem worth while. Not only is it lack luster in the functionality department, but it also fails to be a successful poke like Lee Sin's Sonic Wave / Resonating Strike. I take it at level 4 and max it last, only taking it when it is the only ability available.
As Shyvana's ultimate it is only right that Dragon's Descent be taken whenever it is available. Her ultimate augments all of her abilities in some way. Burnout once again outshines her other abilities, because while in dragon form Shyvana leaves a trail of fire behind her that is very useful in team fights. As it does tons of damage over time to any enemy in it. It is also useful to chase a stealth champion such as Shaco and Twitch when they get low on health and happen to step on it when fleeing to a bush. I have had many stealth champions die while hiding from me in dragon form.

I open with a Doran's Shield, though bland, it gives me the extra health and armor I need to stay in lane early on. I have tried many other builds that recommend Boots and Health Potions as your starting items. But they never seem to be of much benefit. Especially when you have Burnout to catch up to anyone who may be faster and chances are, by the time they have boots, so will you.
By the time you back for the first time you should be able to buy a Vampiric Scepter. This will give a bit more damage output while increasing your sustain. Your next item will be Boots. You may even be able to purchase them at the same time as your Vampiric Scepter.
Giant's Belt will be your next item for purchase, I usually sell my Doran's Sheild about the time I get the Giant's Belt to either get the extra bit of gold to buy it or to buy the Chain Vest if I can get enough gold by selling it.
Follow up with completing your Sunfire Aegis and finishing your Mercury's Treads. It is at this point in the game when Shyvana becomes really threatening. Not only does she have tenacity but she is tanky and fast. You should be able to turret dive with Dragon's Descent to finish of a champ or two and make it out alive with 25% of your health remaining in the most dire circumstance.
The next thing on your shopping list is a Recurve Bow, allowing you to attack faster and deal more damage in less time. Combine it with a Null Magic Mantle to create Wit's End and you will be attacking at break neck speeds with added magic resist. It is around this point that I start tanking turrets that are somewhat low on health, remember that Twin Bite DOES effect turrets. If you activate Burnout when enemy minions are near you should clear them simply because they stand in your AoE.
Follow up Wit's End with Giant's Belt and Phage eventually building them into a Frozen Mallet. You now have even more health and the ability to slow. What more could you ask for? I will tell you. Build an Atma's Impaler for around 70 extra damage every time you use a physical attack. Which, incidentally, you do physical attacks quite often as Shyvana.
Your final item will be a Blade of the Ruined King. It's active will make up a little bit for the loss of Heal and with the added sustain you should be wreaking other bruisers in no time.

Note: There have been times where I have had to build even tankier because our tank isn't doing well or the enemy team just does too much damage. In this instance I found that simply building a Thornmail will cripple champions like Xin Zhao and Fiora.

Jungle Shyvana is an easy concept to grasp for anyone, the main problem is ganking. Without any built in CC Shyvana will have a tough time keeping an enemy in her allies' grasp long enough to secure the kill. To help make up for that, I recommend running Smite Exhaust on her. This will allow you to stop someone from getting away from you.
Shyvana can be jungled in so many different paths it is unbelievable. I personally go the less recommended route and start with Wolves but completely skipping Ancient Golem (Blue). I leave Blue for my second run through the jungle and give it to my mid lane. I move from Wolves over to the Wraith camp. Clear them easily with your Burnout and then move to Lizard Elder (Red). Hit him with Burnout and Twin Bite immediately. You should be able to take his tiny minions out with burnout and get him low enough to Smite. Move to Golem camp then back to heal or gank bottom lane. Your second run through will be easier. Secure Blue for your mid lane. Then gank, ward, or keep jungling till you are needed.
Finally, itemization is almost identical in the jungle. Skip your Doran's Shield and go strait into a Long Sword/ Vampiric Scepter and you are good to go.

In my opinion Shyvana is one of the best farmers in the entire game. All she has to do is activate Burnout and she can instantly farm an entire minion wave in seconds. With all the attack speed she gets from this build she can farm so fast it is almost impossible to keep up with.

Team work is very important as Shyvana. Not only will you be tanking for your team, but you will need to know who to focus and when to initiate. Be VERY careful when initiating with your ultimate into the fog of war. Many a half dragon has been slain because of a poorly timed ult into a bush with the entire enemy team in it. You need to COMMUNICATE with your team. Don't jump into the team if your team says they aren't going in. You can't 1v5.

Pros:

Tanky
AoE Damage
Team Fight Potential
Good Initiation

Cons:

Difficult Ganks
No Built in CC

Flame Breath is Lack Luster

All in all Shyvana is a very solid champion. While she may have flaws, she also has qualities that make her a substantial threat. If you want to live a long time as Shyvana remember that your early game isn't as strong as champions like Cho'Gath so play it safe and don't over extend. Your mid game is solid and you can play some pretty risky moves without worrying too much. But during your late game, you can ult into a team fight and easily come out on top with 30% health or more to spare. Don't forget to use Burnout while Dragon's Descent is active in a team fight. I hope this guide helped. Good luck on the fields of justice summoners.