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I'm doing an electrolysis experiment with an aqueous solution of copper sulfate $\ce{CuSO4}$ and graphite electrodes.

I thought that only metallic copper was formed at the cathode (and oxygen at the anode), but in addition to the copper at the cathode a notable quantity of black substance is formed. It seems to be copper oxide but I'm not sure and I don't know what reaction produces it.

I don't know if it's exactly the same substance, but I noticed that the same black substance is formed by simply immersing a zinc coated paperclip in the same solution of copper sulfate (without electrolysis and without electrodes).

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During electrolysis, the graphite electrode can undergo a process called exfoliation. This happens when the strong electrical field and the chemical environment at the electrode surface cause the layered structure of graphite to loosen. This loosens the individual graphene sheets that make up graphite, resulting in a less compact, more voluminous form with increased surface area. This can appear black and flaky.

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It is the graphite electrode turning into less compact form of graphite. Not really a chemical reaction. Copper oxide cannot form at the cathode.

Copper can appear black, depending on conditions. See:Why does the displacement reaction of zinc in copper(II) sulfate solution result in dark metallic copper?

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