WIP

Smite is essential for any jungler, you can't go without it.
Flash is generally a good pick, as it's always been strong and reliable. Dragon's Descent is a strong escape tool as well so, unlike many other champions, this spell isn't required on her.
Exhaust is an option if you're against heavy damage or champions like Warwick and Hecarim and need more cc. It's not completely necessary, however, because you will likely build Frozen Mallet and if using Glacial Augment you'll already have a slow for chasing.
Ignite is an aggressive choice and although good to inflict Grievous Wounds or finish off a low health enemy, it's not recommended.

Classic Build

First Tree: Precision

Press the Attack: This rune is the best choice on Shyvana, since she uses autoattacks a lot. In 8.5 it received a nice buff and it's now especially good, making it a must have if you're going Precision.
Triumph: Triumph is good for increased survivability in fights and the small gold boost it gives adds up in the long run.
Legend: Alacrity: Attack speed is very good on Shyvana and the other two options are simply lackluster.
Coup de Grace: Simply the best and most reliable rune choice in this situation. You'll likely always have more health that the enemy so Cut Down isn't even an option, and Last Stand is way too situational to be good.

Second Tree: Sorcery

Celerity: More movement speed is always a good thing and it synergizes with Burnout. The tiny bit of extra AD is also good.
Waterwalking: Same as above. This can be replaced with Gathering Storm as it's more situational, but either pick is good.

Glacial Augment Build

First Tree: Inspiration

Glacial Augment: The main reason we take Inspiration. The slow is extremely useful in catching up to your target and sticking to them, especially since Shyvana doesn't have any innate CC.
Magical Footwear: A bit of extra movement speed is nice but this rune is unremarkable and interchangeable with either of the other options, entirely depending on your playstyle. It does save you 300 gold though, so keep that in mind.
Future's Market: I've been taking this rune and liking it, it gives a boost to her already good gold income and lets her get items sooner, which can be insane. Biscuit Delivery is also a good option. Minion Dematerializer is basically useless on her so don't take it.
Cosmic Insight: A little extra CDR doesn't hurt and this rune is just the most useful option we have.

Second Tree: Precision

Triumph: Triumph is good for increased survivability in fights and the small gold boost it gives adds up in the long run.
Legend: Alacrity Attack speed is very good on Shyvana and with the 20% attack speed bonus given by Inspiration+Precision you'll have an insane amount of attack speed.

Stalker's Blade - Bloodrazor: Stalker's Blade is valuable for the slow it gives and attack speed is very nice on Shyvana. Enchantment: Cinderhulk is also a viable option and can also replace Enchantment: Bloodrazor later in the game if your team really needs a frontline.
Titanic Hydra:
Blade of the Ruined King:
Frozen Mallet:
Dead Man's Plate:
Randuin's Omen:
Thornmail:
Gargoyle Stoneplate:
Spirit Visage: A reliable and general magic resist item. It synergizes well with the lifesteal of Blade of the Ruined King and allies' healing effects and is just good all-around.
Adaptive Helm: I personally prefer Spirit Visage just for its reliability, but this item is good against spammy mages and persistent effects like Teemo's Toxic Shot.
Abyssal Mask:

Thanks to jhoijhoi for creating the blue line dividers used at the top and bottom of each section.