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To add to Olin's excellent answer, if you were probing on a digital IO line in order to view a fast changing signal, without the local ground clip (or spring) grounded close to where you make the measurement, you would, in effect, be adding an antenna to that IO line and causing interference to other parts of the circuit and even other circuits in the same vicinity. The signal captured by your o-scope would contain all-manner of unwanted artefacts as well.