I've built this as more of a public garage than a particular guide; both a compilation of data for reference and whatever my current build is. The content herein is prone to change as I rebuild it.

My core is an attempt to reach as close as possible to an attack speed of 2.5/sec. while not sacrificing too much damage mitigation. Until now, I've enjoyed an opening of [Ninja Tabi]+[Wit's End]+[Hextech Gunblade] with Evasion Seals, but, with the loss of the Dodge trait, I've gone to Defense & Shielding Runes, freeing up item slots for more speed.

The numbers are a bit better over on LeagueCraft:

My current style is pure AD. I like Stark's as long as I have Oracle pots; fair's fair.

http://leaguecraft.com/builder/Shyvana/?items=9,21,126,89,89,81&runes=64,64,64,64,64,36,36,36,36,120,120,120,120,120,120,120,120,120,57,57,57,57,57,57,57,57,57,102,102,102&masteries=1300400410013030100000000000000000301400001000000&level=18

I also enjoy the 'Seven Sword Style' AD/AP build.

http://leaguecraft.com/builder/Shyvana/?items=70,108,38,28,44,21&runes=64,64,64,64,64,64,64,64,64,120,120,120,120,120,120,120,120,120,16,16,16,16,16,16,16,16,16,102,102,102&masteries=1030440011000000001304301000310000000000000000000&level=18

In both cases, the Attack Speed bonus from Surge exactly hits 2.5/sec. with a maxed build.

There are those few dragons in Runeterra who have mastered the intense magical energies that course through their unique anatomies, evolving into an elusive Celestial Dragon. These powerful and enigmatic creatures spend most of their time hidden away from the lands of men. However, there are those who find themselves drawn to civilization and who take on human forms to immerse themselves in the world of humans. Occasionally, one finds itself drawn to the spirit of a human and they have relations. Sometimes, in the slimmest of odds, the union will bear fruit and that offspring will be one of the few half-dragons to have graced Runeterra.

Shyvana is one such creature, born to a simple Demacian farm-girl and her Celestial Dragon father. Left to be raised by her mother, her oddities were something that couldn't be hidden from her community, though she was protected by her family. Once her family had perished in disaster, she was left alone for the first time – truly alone. Shyvana quickly became an outsider and it wasn't long before the community's fear began to put her in danger. Facing mortal danger at the business end of an angry mob, she was rescued by the traveling noble, Jarvan Lightshield. The noble was a man who strangely knew exactly what she was and took her under his wing. It wasn't long before she was offered a chance to use her natural skills in the service of Demacia, as Jarvan initiated her as one of the elite guard. Following Jarvan into the League of Legends was only natural.

Shyvana is a fierce warrior, with the blood of one of the most powerful magical predators in all creation flowing through her veins. Unlike some of her fellows, she is a subdued personality – somber, cool, and collected. However, when the moment strikes, her draconian heritage manifests and there are few that can stare her in the eyes and not feel the primal urge to flee.

“By the blood of my father, I will end you!”

Shyvana's abilities don't have any cost related to them, so the use of her abilities is only limited by cooldowns. Instead she uses Fury as her secondary bar.

Fury is only needed as a requirement for the use of her ultimate, which allows her to change form if she has 100 Fury and remain in that state for as long as she has enough Fury. Shyvana won't generate Fury unless she has at least one point in Dragon's Descent, and once she puts the first point on the skill she will fill the bar immediately. Any gained Fury will not be lost until her ultimate is used.

Active: Shyvana next autoattack will strike twice in one swift movement. The second attack will deal physical damage equal to a percentage of her total attack damage. Both attacks will trigger on-hit effects and Fury of the Dragonborn effects.

Dragon Form: Twin Bite will damage all units in front of Shyvana when she uses her next autottack instead of just her target. Each unit hit will be dealt on-hit effects and grant Fury twice.

Fury of the Dragonborn: Each auto attack reduces the cooldown by 0.5 seconds. This doesn't work on structures.

Cooldown: 10 / 9 / 8 / 7 / 6 seconds

Second Strike Damage: 80 / 85 / 90 / 95 / 100 % of attack damage

Active: During the next 3 seconds, Shyvana will deal magic damage each second to nearby enemies and her movement speed will be greatly increased. Shyvana's movement speed bonus itself will be reduced multiplicatively by 15% for every second that passes.

Dragon Form: Shyvana now scorches the earth where she walks, leaving a trail of fire for 5 seconds that will continually deal magic damage to enemies that pass over it.

Cooldown: 12 seconds

Radius of AoE: 325

Fury of the Dragonborn: Each auto attack extends the duration of Burnout by 1 second to a maximum of 6 extra seconds. This works on structures.

Magic Damage Per Second: 25 / 40 / 55 / 70 / 85 (+0.2 per bonus attack damage)

Initial Movement Speed Bonus: 30 / 35 / 40 / 45 / 50 %

Active: Shyvana unleashes a fireball forward in a line that will hit the first enemy it hits, dealing magic damage to it and debuffing them by reducing their armor by 15% for 4 seconds.

Dragon Form: Flame Breath engulfs all units in a cone in front of Shyvana.

Range: 925

Fury of the Dragonborn: Each autoattack that hits debuffed targets will deal 15% of the ability's damage as extra magic damage. (12/18,75/25,5/32,25/39 + 9% of AP)

Cooldown: 12 / 11 / 10 / 9 / 8 seconds

Magic Damage: 80 / 125 / 170 / 215 / 260 (+0.6 per ability power)

Passive: Shyvana reinforces her scales, increasing her armor and magic resistance. These defensive bonuses are doubled while in Dragon Form.

Active: Shyvana transforms into a dragon and dashes to a target location. Enemies along her path take magic damage and are pushed toward her target location.

Range: 1000

This ability has no cooldown but to use it Shyvana needs 100 Fury. Shyvana generates 1 Fury every 1.5 seconds while in Human Form. While in Dragon Form, she will lose 6 fury every second and once the Fury bar is empty she will return to her normal state.

Fury of the Dragonborn: Each of her autottacks will generate 2 Fury. This will allow for Shyvana to prolong the duration of Dragon Form by attacking. This ability works on structures.

Passive Armor and Magic Resistance Bonus: 15 / 20 / 25

Magic Damage: 200 / 300 / 400 (+0.7 per ability power)