As Shyvana, you want to try to deny the enemy jungler with your strong presence early game. Thus, the enemy matchup will directly affect how you play the game
Amumu is weak early on and relies on blue, if for some reason you cannot deny him blue, you should at least take his red. The slower Amumu gets to 6 the better. You can also ward his jungle, take his wraiths when its up. Hell, you can even kill him if you catch him in a bad spot. Get all over this guy, he can't stop the dragon.
Mundo clears the jungle at a super speed and he can start at either red or blue. If you know he started at blue, you should take his red. Mundo without farm is useless, so if you can find a way to shut him down, do so. If he's running exhaust, and you aren't, it will be a close fight, but if you both are running exhaust, you should outright win him. Just don't get kited by Cleavers
Counter with care
Alistar has 2 forms of CC that will keep you in place for god knows how long. However, he neglects his jungle for ganking. Alistar will start blue, get level 2 and then gank mid. If alistar doesn't gank early, then he is doing something wrong. You should take the lead and just take his creeps whenever hes ganking. You know for a fact he'll gank after blue, so pay a visit to his wraiths, and then red. You shouldn't fight him, as his CC will keep you in place long enough for his team to arrive, but taking his jungle is a great way to set him behind.
Gangplank is a weak jungler. He has a slow clear with little sustain. His strength lies in his ability to snowball if fed, and his ultimate. If you can find a gangplank in his jungle, destroy him. Just be careful he doesn't orange your exhaust. He will start blue, so take his red. Without red, GP will have a hard time ganking.
Jarvan is like alistar in the sense that he spends more time ganking than farming. He's a weak 1v1er against bruisers early. You should just walk in and out of his jungle. Although, his CC is potent, so don't engage in a fight unless he's already low, or you might be ganked by his team-mates.
Fight and Counter
Lee is a very weird jungler. He excels at clears, ganks, dueling and counter jungling. You however, are better than him at everything except ganking. You're a stronger duelist (you took exhaust right?) a better cleareer and a godly counter jungler. If you find a lee, fight him. You're stronger than him with your damage output early, that his Q+q combo will not be a threat. If he did take exhaust, be VERY careful, its a close fight. Lee's will start at red and gank, so taking his wolves->blue is a good option, but warding his jungle is a better option. Setting Lee behind is probably the best thing you can do for your team.
Counter with care
Maokai is like Alistar. Buttloads of CC. He can deal some nice damage early with his combo, but his CC is what's the problem. Counter his wraiths, take his red, disrupt his early set-up and you've successfully held Maokai back. Be warned, his ganks are strong, so careful warding is advised.
Counter with care
Like maokai, he has a lot of CC, but most of his damage comes from his shield, which you burst through in mere seconds. His early clear is slow, so take his red since he MUST start blue. You can fight him if he's low, but his array of CC will make it so that their team will be a problem. Also, he can anchor to walls for an easy escape.
Do NOT fight, countering okay
Nocturnes Q adds a lot to his damage output, and he can out damage you and fear you if you aren't careful. Do NOT fight a nocturne unless he is low. If you took exhaust, you can win the fight, just be careful, if he has taken exhaust he will come out on top. Taking his red is a good way to set a nocturne back.
Like all other gank oriented junglers, Rammus has a weak clear. Taking his wraiths and red is a good way to set him back, because Rammus needs level 3 to gank, delaying it will make your lanes more prepared and/or have the ability to stop him if he comes for a gank. His CC is a problem, so don't get too overzealous, but you will win in a straight fight.
Her clear is slow, she deals no damage and has no sustain. Her CC is okay pre-6, but nothing special. Destroy this girl. Take her red, take her wraiths, take her lunch money and make her cry.
Fight him with care
Shaco is a trickster, and thus will try to make your life hell. If he's taken exhaust, be careful about the trap placement and you should come out on top. Against a shaco, I always reccomend starting cloth and 5 pots so he can't catch you low on HP.
Fight him with care
Skarner is a beast. He jungles fast, stays at high HP, and has a great slow that will keep you around for long periods of time. If he's low, you should totally kill him with exhaust, but if he isn't, leave him. He will kite you around long enough for his team to arrive.
Fight with care
Trundle saps AD, has a pillar that blocks paths, a steroid, and a good clear time. Do NOT fight trundle.
Counter but do not fight
You can take a phoenix udyr in a fight, but a tiger will shit on you. You can try and counter him by warding his jungler, but he is a fast clearer. He is similar to mundo, so treat him that way.
Destroy him. His sustain may be high, but he has NO dueling power early on. His clear is also slow. Take his stuff, make him cry, make him feel like a Sejuani. Delay his 6 as much as possible and you've won the jungle.