Depends. A spell that targets an object should mention how it works in the spell description.
As a default, attended objects use the bearer's saving throw or it's own, whichever is better.
Did you have a particular spell in mind?
Keep in mind that acid doesn't necessarily bypass hardness, and the CRB gives few guidelines for deciding what objects are "weak against" particular energy types.
I don't know if you can use attack roll spells against attended objects. Normally you'd have to use sunder, but there's no ranged sunder apart from a single archetype, and the fact that you're using a spell doesn't really change that--again, unless the particular spell allows it.
There are three possibilities to attack attended items.
1) Sunder combat maneuver with a melee weapon or with some touch spells.
2) Target of a spell rolls a natural 1 on his saving throw against a damage spell like fireball.
3) Special spells like shatter.
Maaayyyybee (i dont know) there is something out there that let you perform combat maneuvers with range attacks like the 'Trick Shot (Bow only)' ability from the fighter archtype 'Archer'. If you find something than you can sunder with rays.
The very idea that a spell which allows an attack roll is somehow magically incapable of targeting someone's sword is anathema to the spirit of the rules.
There's seriously no rules in place for this? That's... really weird.
Depends on one's idea of the spirit of the rules. Wizards generally cry foul if the enemy sorcerer scorching rays the wizard's spell component pouch. So the designers made it very difficult to damage held objects without (a) being in melee or (b) using spells specifically designed to target objects. There are quite a few of these; warp wood is one of my favorites.
Meh, you made it to melee without the enemy buffing/debuffing to crazytown. The cleric had it coming.