As a general guideline, it is preferable to connect any cable shields to the metal chassis (not PCB ground), at just one end. To keep things simple when multiple shielded cables connect out from some central device, the shield connection should be done only at the "hub" device, and left open at the "spoke" devices.
Similarly, for a chain of devices connected by shielded cable, each shield ought to be connected to the chassis of the upstream device, and left open at the downstream end. Yes, this does mean that the shields of different links of such a chain might be at different potentials, depending on how well earthed the individual devices are, but this is generally not a problem.
A basic, inexpensive addition incorporated in many consumer device cables today, is a clamp-on ferrite bead or "split bead" at each end of the cable, close to the connector. This reduces high frequency EMI off the shield with minimal complexity. A good document about such RFI beads is here.
Note that the chassis is usually connected to "earth" of the location, not to ground of your circuit board.
Of course, this is a simplification of a fairly complex subject, but it serves the purpose for designs at frequencies where surface effects do not predominate. As frequency of signals involved rises into GHz, other factors need addressing.
It's worth having a USB cable with ferrite 'stoppers' on it, to attenuate conducted emissions along the cable. This will reduce the chance of hash from the switch mode power supply and other PC generated interference 'getting into' the audio circuits of the ADC that's doing the recording. With an electrically noisy PC, and a cheap ADC, a cable without stoppers might produce breakthrough interference on audio. However, part of the cost of a good ADC should be high immunity to conducted interference.
The ferrite stoppers add a few pence (cents) to the cost of a cable, and most cables I've seen have them (the cylindrical lump close to the connector at one or both ends of the cable), only the very cheapest cables do not.
Once a cable has these stoppers, all cables either work, or are broken, there are no differences in data quality. If it can transfer a file, then it can transfer your highest quality recording. Golden ears cannot tell the difference between two identical files.
I wish I could dream up snake oil like expensive 'audio grade' USB cables and sell them for a fortune to audiophools, but I'm just an honest engineer!
Best Answer
A shield is used to prevent radio waves from coupling into signal or power wires within a cable.