It looks like you have a stash of 0.1uF caps available. Put two of them across input/ground at your 7812. Put another one across 12v out and ground right next to the 7812. The transients at the NE555 output should go away.

 

In my opinion, I think the most important place for the capacitor is right across the supply pins of the mosfet driver. That overshoot is a result of the inductance of the leads that power it (rail and ground path), so decoupling needs to be done at the chip itself. Also, unless you are using surface mount capacitors, you will need more than one capacitor. Those electrolytic caps have long leads, and most electrolytics are made by rolling a long piece of foil rather than paralleling several short foils, so that added inductance reduces its effectiveness at higher frequencies (which you have with that fast edge!). Surface mount capacitors have short leads regardless of value, so a single .1uF chip would be ok. But if you are using leaded components, you will need a .1uF electrolytic and a couple smaller value ceramics (like 1nF and 10pF) with the leads as short as possible. You need to put these right at the chip's power pins!

 

I still want to know why this very minor overshoot is a problem.
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I doubt it is a problem of functionality, rather one of curiosity. I used to try chasing down little problems like that when I first got into the hobby more so to understand them as it was a deviation from the theoretical examples given. Introductory books on just about any subject, be it electronics, chemistry, etc. do a good job at giving people confidence in a particular matter, so when things are not as expected, it challenges that confidence. I would want to know why even if the problem wasn't a hindrance to functionality. He was on the right track questioning ground loops, and knowing that term alone makes his passion for the hobby evident. To that end, it doesn't matter if it affects the circuit problem, it's an intellectual problem that needed solving!

 

I'm embarrassed to say that I didn't think of it from that perspective.

My general "Shot-Gun-Approach" solution is that if it's not already extremely Low-Impedance,
or there's any chance of noise getting into the "good" Signal,
then it needs a Schmitt-Trigger for a receiver, or an Op-Amp, or a Comparitor, with Positive-Feedback.
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