|Bob Bob Bob|
That's... actually a kind of messy issue. We have a FAQ, but many disagreed. A grapple is done with the hands (that's why the penalty for not having 2 hands free), but it doesn't benefit from anything that enhances your unarmed strikes (which many people think of as punches). Ditto for Bull Rush, Overrun (stomping), and Steal. And Dirty Trick if the hand is incidental (throwing sand in their eyes) but not if it's not (eye poke). With nothing modifying it, the simple answer is "Disarm, Dirty Trick, Sunder, and Trip (and therefore Drag and Reposition) can be done with weapons". You aren't required to use a weapon except for Sunder, and Dirty Trick requires a description of how you're using a weapon before it counts (DM discretion).
Then we get into the ways you can make a maneuver with a specific weapon. Shield Slam for Bull Rush with your shield, Tetori Grab on unarmed strikes for grapple with an unarmed strike, and I think some feats for Overrun and an attack at the same time? It's complicated. The long answer is "without a specific rule saying you can do something different the only maneuvers that allow a weapon are Disarm, Dirty Trick, Sunder, Trip, Reposition, and Drag, with the following caveats: Dirty Trick requires the DM to approve of the use of the weapon and Reposition and Drag require the weapon to have the Trip property".
That's just for weapons. Other bonuses fall under "When you attempt to perform a combat maneuver, make an attack roll and add your CMB in place of your normal attack bonus. Add any bonuses you currently have on attack rolls due to spells, feats, and other effects." So this includes Inspire Courage, True Strike, and anything else that's "+X to attack rolls". Weapon Focus applies if you're using a weapon (and Weapon Focus (grapple) can be taken). There, more information than you ever needed.
Yeah, combat maneuvers get weird. And Bob didn't even mention some of the other things, like a Dang Bong, which grants a +2 Bonus to Grapple...unfortunately, you take a -4 penalty for not having two hands free, so unless you're dual-wielding Dang Bongs, you end up still grappling at a minus penalty.
|Bob Bob Bob|
Oh no, it's apparently even weirder than that. You can use grappling weapons to grapple... provided they crit.
Grapple: On a successful critical hit with a weapon of this type, you can grapple the target of the attack. The wielder can then attempt a combat maneuver check to grapple his opponent as a free action. This grapple attempt does not provoke an attack of opportunity from the creature you are attempting to grapple if that creature is not threatening you. While you grapple the target with a grappling weapon, you can only move or damage the creature on your turn. You are still considered grappled, though you do not have to be adjacent to the creature to continue the grapple. If you move far enough away to be out of the weapon’s reach, you end the grapple with that action.
That's not even addressing how this works with the Mancatcher (which has Reach) and "If you successfully grapple a creature that is not adjacent to you, move that creature to an adjacent open space (if no space is available, your grapple fails)." Which would immediately end the grapple, as they're now out of the weapon's reach. It's... complicated. Talk to your DM beforehand.