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How high is the density of asteroids in the rings of Saturn?

Would it be possible for a spacecraft to fly through the thicker rings without colliding with asteroids? Would it be necessary to avoid a collision through careful maneuvering between the larger and smaller rocks? Or would it actually be a coincidence to even see an asteroid when flying through the rings?

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No, you can't fly through the rings you can see without hitting lots and lots of dust-sized ice particles at high velocity. Your vehicle will not fare well. The ring material is not sparse in that sense.

There is a wide distribution of particle sizes from boulders to dust. There's a lot more dust-sized, so that's what you need to worry about. (I wouldn't really call the particles "asteroids". They're almost entirely water ice.)

You shouldn't even fly through what appear to be gaps without at least looking at the occultation data and doing a calculation for what you might hit.

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