The spell would need to affect the area around the caster for that to leave an aura behind, or they would need to cast spells wherever they passed. What you want seems more in the realms of ranger spells, or even inquisitor spells.
The lingering aura of a Fireball could be detected on the area, but not spells that were cast on the caster, or on another person. Unless those are the targets being analized. Or if the caster did cast something like Trail of the Rose, that affects an area where the caster passes.
If the spell effect has a Duration, it can be detected as long as the spell lasts. But if the spell effect was Instantaneous, it's Lingering Aura would last only for a short duration, and could be seen with Detect Magic.
Lingering Aura: A magical aura lingers after its original source dissipates (in the case of a spell) or is destroyed (in the case of a magic item). If detect magic is cast and directed at such a location, the spell indicates an aura strength of dim (even weaker than a faint aura). How long the aura lingers at this dim level depends on its original power:
Faint 1d6 rounds
Moderate 1d6 minutes
Strong 1d6x10 minutes
Overwhelming 1d6 days
Invisibility, being a 2nd level spell, would generate a Faint Aura, and lasts only for 1d6 rounds after being dissipated.
But Greater Invisibility, a 4th level spell, would last for 1d6 minutes.
A petrification effect would depend on what caused this effect, most creatures's supernatural petrification works like Flesh to Stone, which is a 6th level spell, but has an Instantaneous duration, thus it's aura would last only for 1d6 rounds as a Moderate Aura.
- Cockatrice: as Flesh to Stone;
- Basilisk: as Flesh to Stone;
- Medusa: Does not say, which means it can be detected as a magical aura (being a supernatural ability), but you cannot detect the strength of this aura or identify it. And it most likely does not leave a Lingering Aura behind.
That is, if the creature had no magical items on her, and no spell with a duration. Otherwise, their auras can be detected normally with detect magic or arcane sight. You will see that nothing on Stone to Flesh states that the auras disapear or go inert, which means they are all active regardless of the state of their target.
If a 10th level mage was petrified with Mage Armor (1 hour/level duration) cast on him, his aura could be detected for 10 hours, plus 1d6 rounds.
Similarly, if he had a Cloak of Resistance +1, anyone would detect that as a weak transmuration aura.
There are better options to find someone you want to follow using magic though, such as the spell Locate Creature.