If a structure with delocalized electrons participates in a reaction, which resonating structure do you use to write down the mechanism? Is using either equivalent? Is it preferred to use the resonance hybrid? If so, what are the rules of using curved arrows for partial bonds?
Any resonance structure can be used. Why? Resonance structures represent multiple views of the same compound, so it must be possible to start with any valid resonance structure. Because resonance structures consider different positions of electrons, the sets of arrows needed to describe the mechanisms of different resonance structures will be different.
While it's possible to use any resonance structure in a mechanism, selecting one resonance structure can be used to improve communication or emphasize particular property/reactivity of a compound.
As an example, below is a figure describing the formation of an enolate and its attack on an aldehyde (aldol reaction) using different resonance structures. For clarity, all of the curved arrows are describing reactions, not the relationships between resonance structures.
We know that carbonyls such as 1 have an important resonance structure which places a negative charge on oxygen and leaves a positive charge on carbon (2). While 1 is the most important contributor, 2 emphasizes the carbocationic nature of the carbonyl carbon. I like this structure, because I know some reactions of carbocations: SN1 and E1. This helps explain to me why the alpha-position is acidic. Deprotonation of 2 next to the carbocation, just as in E1, gives enolate 3. Thinking of the carbon-oxygen pi-bond as a leaving group, deprotonating 1 at the alpha- position gives the same enolate 3 but by a different set of arrows compared to 2 to 3.
Note that there is a third way (not shown) to describe the same deprotonation where a base deprotonates the alpha-hydrogen and the electrons are simply localized on the alpha-carbon to give structure 4.
Structures 3 and 4 are resonance structures, and again these can both be used to describe further chemistry. The blue set of curved arrows from 3 to 5 contains one additional electron pair movement to the red set from 4 to 5. Both are valid descriptions of the attack on an aldehyde, but emphasize different aspects of enolate reactivity.
Keep in mind that bond-line drawings, hybrid drawings, and curved arrows are just simple representations of molecules. Thankfully they work very well and allow us to reduce the complexity of electronic structure to a manageable level.
Finally, as @orthocresol suggests in the comments, using hybrid drawings is difficult. This is because bond-line drawings represent electron pairs as localized and curved arrows work best when showing the action of electron pairs.