present a flux vector model for the interaction of spins with thin-film superconducting wires, and show how measurements of flux as a function of the direction of an external magnetic field applied in the plane of the wires can reveal the value of impurity spin quantum number and the nature of its interaction with the circuit. We describe a method to accurately calculate the flux produced by spin impurities in realistic superconducting thin-film wires, and show that the flux produced by each spin is much larger than anticipated by former calculations. Remarkably, flux noise power due to electron spins at the thin edge surface of the wires is found to be of similar magnitude as the one at the wide top surface. In addition, flux noise due to lattice nuclear spins in the bulk of the wires is found to be approximately 5% of the total noise power. We discuss the relative importance of electron and nuclear spin species in determining flux noise, and propose strategies to design low noise SQUIDs.