Checking the rules for the petrified condition, Enlarge Person and for transmutation spells in general shows that there is nothing that mutually excludes both effects at once. Specifically, a petrified humanoid does not change type and become an "object" - it is still the creature, but with a condition applied.
There is no apparent "freezing" of ongoing effects, spell durations etc, which should really be noted under the condition if that was the case. There may even be undesirable exploits to make cheap permanent effects tied to petrified creatures under otherwise temporary effects.
So in my opinion, that supports your option 2:
(2) The petrification creates a statue of one's current form, but when
the spell wears off the new stone statue changes shape.
However, I think there is ample room for DM interpretation supporting either of the other two options, and even variations depending on what kind of effects can be "frozen in" when a character is petrified. It would be a minor ruling either way, and only require a little consistency during the game.
A side-effect of supporting option 2, is that if you cast any visible effect that does not affect normal objects on apparent statues, then you will see which ones are in fact petrified creatures from how the spell works (or not). As there is no competing "detect petrification" spell as far as I know, I think that is probably ok.
Not-Unconscious does not necessarily mean awake and aware.
Sometimes the game just expects us to understand certain terms without giving explicit rules text about what they mean. Famously, 5e doesn't specify a 'dead' condition or give any rules about what it means to be dead. So there is a precedent for the game having certain game concepts left to the players' understanding of the world without giving specific rules text.
The Unconscious condition has specific effects that don't jibe with petrification: Unconscious creatures fall prone, drop their belongings, and take automatic critical hits, none of which makes a lot of sense with a creature being turned to stone. So there's a good reason not to bring those into a petrification situation.
Which is a all a rather long way to say 'the DM decides that', but:
What rules text even represents being mentally shut down?
Other than the actual name of the condition, the "incapacitated, can't move or speak, and unaware of surroundings" bullet point is as close as the Unconscious condition ever gets to specifying that your mind is shut off, and that text is present in Petrified as well.
So, for whatever it's worth, it seems like Petrified includes the mental component of unconsciousness, in so far as that state is represented in rules text at all.
Best Answer
The flesh to stone spell has instantaneous duration. Thus there is no “normal” aura after the spell is cast. But it is generally agreed (I can't find unambiguous RAW entry for that) that since magic is present at the time of casting, the spell leaves lingering aura as normal. The flesh to stone spell is level 6. It results in moderate aura at the time of casting and a lingering aura persisting for 1d6 minutes (see detect magic). After that no aura can be detected.
Your option after the time has passed is the true seeing spell.
Other similar spells or spells specifically designed to detect creatures turned to stone may exist in other sources that I don't know. Alternatively you can work on researching such a spell with your DM if he/she allows spell research.
At the same time spells are not the only way to tell petrified creature from a statue.
In D&D 3.5 Spellcraft skill can be used to identify materials made by spells.
Pathfinder has this option for Knowledge (Arcana) skill:
Finally, you can undertake a regular investigation.
For example, you can learn what clothes the prince was wearing before disappearing, or you can find out what statues of said prince were ever made (if any) and where they are situated. Based on this knowledge you can make a decision on whether this statue is a regular statue or a petrified prince.
People in the comments also note that a statue will bear marks of the instruments used to sculpt it, thus a Profession (Sculptor) check may help to discover them.