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The performance of SQUIDs and superconducting qubits based on magnetic flux is degraded by the presence of magnetic flux noise with a spectral density scaling approximately inversely with frequency. It is generally accepted that the noise arises from the random reversal of spins on the surface of the superconductors. We introduce a numerical method of calculating the mean square flux noise (&PHgr;²) from independently fluctuating spins on the surface of thin-film loops of arbitrary geometry. By reciprocity, (&PHgr;²) is proportional to (B(r)²), where B(r) is the magnetic field generated by a circulating current around the loop and r varies over the loop surface. By discretizing the loop nonuniformly, we efficiently and accurately compute the current distribution and resulting magnetic field, which may vary rapidly across the loop. We use this method to compute (&PHgr;²) in a number of scenarios in which we systematically vary physical parameters of the loop. We compare our simulations to an earlier analytic result predicting that (&PHgr;²) R/W in the limit where the loop radius R is much greater than the linewidth W. We further show that the previously neglected contribution of edge spins to (&PHgr;²) is significant--even dominant--in narrow-linewidth loops. [ABSTRACT FROM AUTHOR]