All those fly wires contribute ballpark 1nH per mm of length. Doesn't sound like much, but it adds up.
Inductance means a voltage drop when current is changing (V = L dI/dt). You didn't mention what transistors, or what load current, but I'm guessing some amperes, and it can probably be switched in ~100ns or less, so wire lengths of ~100mm might be dropping several volts peak.
I think you will also find, the noise is visible on "ground" itself (i.e., probing the ground clip where it's connecting to the circuit), or pretty much everywhere else in the system. This is common-mode noise. It means there is voltage generated with respect to some other ground, making a loop back to your oscilloscope (through its power cord ground). It might be the power supply, PC (via MCU + USB connections), or straight-up radiated to space (long wires to power supply / motor?).
All together: it seems you have voltage drops, due to changing current, across wires including what is notionally "ground". In quotes, because there's the fiction of what we want ground to be (an ideal zero-potential reference), and there's the reality of what happens in circuit (voltage drops everywhere).
The best way to avoid this, is to trace current loops in the system: notice the gate-source loops from the drivers, the source-VDD loops from the bootstrap diodes (D2, etc.), the +V/out/GND loops from each inverter -- minimize each of these individually, with shortest wiring lengths, small loop areas, and wide conductors (broad facing area with narrow spacing: think wide copper pours on top and bottom of a PCB).
Indeed, the best single design principle you can employ, is the ground plane.
The best way to do this for a proto, is to use copper-clad PCB, and either cut pads out of the top layer, or add pads on top ("Manhattan" style) to make connections over the top copper (which will be your plane). The plane is a very wide conductor -- still not ideal, but a vast improvement over loose wires -- and its proximity to your conductors keeps loop areas small. Most of all, it acts as shielding, in that the magnetic field from one wire is mostly blocked by the plane, rather than shared by nearby wires (mutual inductance, coupling).
Which is another big problem with the style of layout you've used: the wires are wide open, magnetic fields go everywhere. Hence, you pick up those transients pretty much anywhere in the build -- or indeed in the space above it (try clipping probe tip to clip, making a one-turn loop, and waving that over the circuit). With ground plane design, you'll find that goes away, and you have to actually clip into the circuit to see nasties -- and what nasties will be there, will be much smaller!
As for the circuit design, mind that you have outside signals coming straight into comparators. LM339 responds at up to a few MHz. You may be having chatter here, due to the high frequency noise; maybe this doesn't do anything (latched receiver?), or maybe it's disastrous (those signals being labeled INTR (interrupt?) is a little unsettling..). In short: consider adding at least an RC filter to the motor sense lines, and maybe hysteresis as well.
