Arduino Motor Shield R3 Current sensing
From the block diagram in the datasheet, you can see that the current sensing is influenced by the output stages of the H-bridge. This means that when you use PWM to drive the motor, the Sense voltage will vary with the PWM signal. The voltage across the sense resistor is proportional with the current through it, which in turn depends on the voltage across the motor's induction. During the positive period of the PWM signal, the current will be 'slowly' increasing whereas during the negative period of the PWM signal the current will be 'slowly' decreasing. In practice this will look a bit like a triangular wave and that is the noise you measure.

The solution is to make sure that you synchronize the sense voltage measurement with the PWM signal. For example you can attach an interrupt to the timer that is used for the PWM signal and it requires a fair bit of digging into the microcontroller's datasheet (and some trial-and-error) to figure out how to realize that. Another option is to connect an interrupt pin to the PWM output pin, which may be simpler to start with. It can be done, though you might need to lower the PWM frequency (to fit the ADC conversion within the PWM duty cycle), directly program the hardware registers (Arduino libraries are slow) or use some assembly.