Your problem isn't so much the scope probes as the breadboard.
The breadboard adds series inductance (as though you had installed coils in series with the SPI signals) and parallel capacitance (as though you had put small capacitors from the SPI signals to ground.) This will slow the rising and falling edges, making it more difficult to detect level changes. The inductance can also cause ringing - there can be false level transitions from the ringing.
Attach your probes directly to the SPI signals, right at the point where the wires connect to one of your boards.
Use 10X probes if you have them - they are less of a load on the signal lines.
SPI uses relatively fast signals, and isn't intended to be used between circuit boards.
It is intended for use on a single board with short connections.
The wires between the boards are probably making your SPI signals "unhappy," never mind the breadboard and the additional load of the oscilloscope.
- Use short wires.
- Keep the circuit neat.
- Use straight wires rather than letting things twist around and get tangled.
Put the breadboard down and back away slowly. Nobody has to be injured here.