Hello there, I'm betting you came here to see a rapeface, AD/AS, on-hit effect, Shyvana build. If that's true I'm sorry to disappoint you. This build actually focuses on AS/AP, and the only on-hit effect is that proc'ed by her E. That being said I've had se success with this build, and find it pretty fun. So hopefully you will try it out before downvoting me for going AP Shyvana.

PS: This is my first build and also a fairly rough draft, I am hoping to get some feedback before I start to put a ton of time into this guide. So you will find some spelling and grammatical errors, feel free to point them out so I can fix them, and feel free to give me any suggestion you might have. Please be gentle, hehe.

PSS: For any who don't know, the stats at the very top of the build guide don't take into account items passives, such as that from Rabadon's and Guinsoo's, your AP is actually just under 500, your AS about 1.50 (with your Guinsoo's stacks).

Pros:
-Extremely fast jungler
-Tons of AoE burst damage (and AoE slow from Rylai's)
-Decent sustained AoE damage even after CD's are blown
-Able to focus down single opponents through proper use of your E
-Best mix of AoE burst damage, sustained damage, and 1v1 focus damage
-Fairly high survivability
-Your ultimate is easy to charge with your autoattacks
-She's a freaking dragon!

Cons:
-Is NEVER a good fit for Dominion
-Not the tankiest Shyvana build
-Requires some skill to make sure your burst actually hits your target
-Is somewhat reliant on your team, not the most effective 1v1 build
-Her ganks are as bad as Olaf's, lol

Mercury's Treads:
These boots are non-negotiable, they are your only MR item, and the tenacity can quite literally be the difference between a win and a loss.

Rylai's:
Very important for it's utility and survivability, with this you can Ulti the enemies towards your team and slow them. This can allow you to split off the squishes and force them into your team allowing you to pick them off easily (supposing you land your Ulti in the right spot). Also, your W and E will both slow, so basically you can perma-slow the enemy team by 15%. Then in 1v1, in human form your E nuke will slow for 35% making it very easy for you to reach your target, get you W on him and proc that passive.

Rabadon's:
This build focuses on damage from your E and R which both scale with AP, so getting this is a no brainer.

Guinsoo's: AD, AS, and AP! What's not to like about this item? This means more procs, more nuke, and more damage on your W.

Sunfire: Damaging aura, armor, and more health. This helps make sure you can dish out continual damage even with your E and R down, also helps keep you alive.

Gunblade: AD, AP, lifesteal, and spell vamp. This will constantly replenish your life as you stay in the heat of battle. Which means less going back for health at base.

Marks:
Took attack speed runes, these will help you charge your fury, letting you get your ultimate up more often. Also, your passive causes you to hit targets marked with your E for 15% of it's damage per hit. In any 1v1 it is imperative that you hit with your E (slowing and debuffing your target) and proc your passive as much as possible. Your E's total damage with the full build should be about 600, meaning each autoattack will hit for 195 (your attack damage) plus 90 (from your E). You should get off eight autoattacks in the time the debuff lasts (four normal autoattacks, and two doubles with your Q). That adds up to a nice 2,280 damage over four seconds (not including damage from your W, sunfire cape, ulti, or even the nuke from your E).

Seals:
Refer to marks.

Glyphs:
Shyvana's main resistance weakness is her MR, to fill that I take scaling MR/lvl runes for survivability.

Quints:
The utility is pretty much worthless to Shyvana (besides experience gain and buff duration of course), so I see no point in wasting 9 points going into it. Instead I take Quints for experience, which actually are a little more effective than the experience mastery.

For masteries I go 9/21/0:

Offense:
I grab smite obviously, and AP/lvl, CDR (this is the only CDR in this build), and MR pen to help your R/E AoE nuke hit the enemy team for more.

Defense:
I grab armor and MR (armor because I don't take cloth armor for my jungle route). The big one here is Ardor giving +4% AS and AP, which are the two core damagers in this build. Then of course I finish off with 4% damage reduction to make my build a little beefier.

First I take a point in W as it is your main tool in the jungle, putting out a consistent damaging aura which will be draining the neutral creeps of health quite quickly. Then I take a point in Q, followed by a point in E. Your Q let's you burst through the wraiths like nothing, so at least on level in that is required. I then max W first for the speed boost and damaging aura, followed by E for it's nuke and the 15% damage proc, finally maxing W last. Obviously take a point in your Ulti whenever it is available.

Q - Twin Bite:
Resets your auto-attack timer and causes you to hit twice with your next attack. This procs on-hit effects (most notably with this build the on-hit effect with your E), the second attack only does a portion of your attack damage until you get it maxed though. I mainly use this skill to proc my E since I don't build any real AD or on-hit effect items, without E it's damage is negligible. In dragon form this hits all units in front of you for a measly range and does not proc on-hit effects as far as I know. Also, this skills CD is reduced by 0.5 with every auto-attack.

W - Burnout:
Basically acts as a Sunfire Cape on steroids. It puts out a damaging aura which scales with AD. This also makes you run at an increased speed for it's duration. This is your main creep farming skill, it also aids in chasing, escaping, and ganking. With Rylai's it slows enemies hit by 15%. In Dragon Firm this leaves a fire trail behind you with the same effect as the main skill, and your passive makes it last longer (up to six seconds) adding one second for each autoattack you land.

E - Flame Breath:
A semi-long range nuke, it hits for 260(+0.6AP), which is about 600 damage with this build. This skill also marks those hit with a 15% armor reduction debuff, and those Shyvana lands auto-attacks on marked with it suffer extra damage equal to 15% of the skill's damage (~90). In Dragon Form this becomes and AoE nuke. It is important to not only note the damage but utility of this skill as well. The armor reduction makes it easy for your carries to rip apart even the tankiest team, and with Rylai's it applies a slow to those hit. This is also your main damager in 1v1 combat thanks to the nuke and on-hit effect.

R - Dragon Form:
You transform into a dragon and fly to your target location knocking enemies hit towards that location and damaging them. This is part of your AP nuke doing 400(+0.7AP), about 800 damage, to an area. While in Dragon Form all your basic abilities are modified as outlined in each abilities description. This also passively gives you armor and MR, which double when in Dragon Form. This does not run on a CD timer, Shyvana gains fury over time, or by using auto-attacks. It is important to keep your fury charged AT ALL TIMES! This is easily done by autoattacking a minion blob to death, or a quick run through the jungle. This skill has a ton of potential uses: ganking, escaping, chasing, nuking. You can use it to push carries away from tanks, or tanks away from carries. It is effective in map control, as well as pure damage. If you get yanked in the enemy jungle, Ulti away. If someone escapes from you with low health, Ulti right onto them. If someone tries to escape your team, Ulti them back towards them. You can use this in almost any situation, and especially in team fights, your damage output rockets with this and it's modification to your W and E.

Summoner spells I use:

Smite: OBVIOUS, lol.

Flash: Can flash into position for your Ultimate to push enemies towards your team. Or in dire situations to escape.

Other viable spells:

Cleanse: You will get CC'ed fairly often so this will see some use.

Ghost: Doesn't have the surprise factor of flash but will allow you to cover much more distance.

Surge: When this comes out I think it may be viable to take this instead of Smite, is the stats are anything like Rally now you should be able to use it to help you burst through monsters without the use of Smite. Since Smite is useless in teamfights, this would be of more use lategame (assuming it does what I think it will).

I take the standard golems, wraiths, wolves path. I start off with a boots and 3 health pots, get a pull on the double golems (make sure they don't leech exp), finish them both out without using smite, grab a level in Q. Smite the big wraith, run into the middle of the small ones, W, and then spam Q. Your basic combo in the jungle will be to keep your W up in the middle of the creeps while spamming Q. After finishing wolves go buy a vampiric scepter and health pots (if you have extra gold). Then I start back at double golems making a clean sweep over to blue, I save Smite to finish off the red and blue buffs. If you have any health pots I usually try and steal the enemies golems. If they catch you use either your Ulti or Flash to escape. After level six the fun begins, if your lanes have CC let them initiate then Ulti in. If not Ulti in behind the enemy to surprise them, unleash your E on them then your W, autoattack and spam Q as it comes up. After you get Rylai's (usually rushed after vamp scepter) ganking becomes infinitely easier.