A surprise spell must be a targeted spell
The arcane trickster special ability surprise spells, in part, says, "At 10th level, an arcane trickster can add her sneak attack damage to any spell that deals damage, if the targets are flat-footed." Unlike, for example, the spell magic missile that has the entry Targets: Up to five creatures, no two of which can be more than 15 ft. apart, the spells black tentacles and fireball have no Target entries so they have no targets (and, in fact, most area spells don't). Employing a targeted spell is one of the requirements for the special ability surprise spells to trigger.
Merely being invisible is insufficient to render a foe flat-footed against the invisible creature's attacks. The invisible attacking creature does gain a +2 bonus on its attack rolls and the foe does lose it's Dexterity bonus to AC against the attack, but the foe is not actually considered flat-footed against attacks launched by the invisible creature, therefore the special ability surprise spells won't trigger by virtue of invisibility alone.
So to benefit from the special ability surprise spells, the arcane trickster can, for example, on the first round of combat cast a spell that has a Target entry at typical foes (like those without the extraordinary ability uncanny dodge) that have yet to act during that round. In such circumstances, the arcane trickster's special ability surprise spells usually triggers.
…And Paizo disagrees
Thanks to Tyelor K I learned that the FAQ includes this exchange:
How does the Surprise Spells class feature of the Arcane Trickster prestige class… work with spells like magic missile and fireball?
The Surprise Spells class feature allows the Arcane Trickster to add his sneak attack dice to spells that deal damage that target flat-footed foes. This damage is only applied once per spell. In the case of fireball this means it affects all targets in the area, with each getting a save to halve the damage (including the sneak attack damage). In the case of magic missile, the extra damage is only added once to one missile, chosen by the caster when the spell is cast.
Note that this exchange agrees that the "Surprise Spells class feature allows the Arcane Trickster to add his sneak attack dice to spells that deal damage that target flat-footed foes" (emphasis mine), yet then this exchange goes on to discuss fireball as if it were a targeted spell, even though it's not. If going by the FAQ, then, a player needs to ask the GM if each individual spell is eligible for surprise spells or not because target in that FAQ entry means something different from the game's normal definition of target.