Shyvana has long-ish cooldowns early and she doesn't have any innate sustain. This result in her being vulnerable to being counter-jungled at her first or second buff; while she's a good duelist, you should be able to beat her if you catch her with her abilities on cooldown. This is further exacerbated if the Shyvana takes Exhaust instead of Flash.
Shyvana has to be in melee-range to deal any significant damage, so she can be susceptible to peel and CC. Displacement abilities are especially annoying since Shyvana's only really gapcloser is running at her target (with the exception of her ultimate, but they will usually lead with it in teamfights).
If Shyvana builds Devourer and Blade of the Ruined King early, she'll have high damage but she'll be squishy and easily bursted in the mid-game; she'll be a terror in duels but less of an issue in teamfights if focused. It can be a bit more difficult to stop her if Shyvana gets a defensive item (i.e. Sunfire Cape, Randuin's Omen, Banshee's Veil) after Devourer and before Blade of the Ruined King.
Shyvana is pretty weak at ganking lanes early in the game. It's not impossible for her (like most junglers, she can easily gank an over-extended lane, especially if they lack an escape and/or wards), but it is much more difficult. Gank-oriented champion can take this time to snowball their allies and put them ahead of the enemy laners.
In my experience, a good Nunu is probably one of the worst opponents for Shyvana to face. In the early game, he can control buffs, Rift Scuttlers, and Dragons to deny Sated Devourer and vision control. Later in the game, Nunu can prevent you from sticking to opponents with his Iceball and ultimate while Bloodboiling his ADC to increase their DPS and mobility. While Nunu can't duel Shyvana, he can easily escape her or chase her down with W and E and there isn't much she can do to stop him. Nunu can easily build Sightstone for track Shyvana throughout her jungle, preventing her from ganking and letting Nunu know where he can counter-jungle safely.