As a means of dynamically reconfiguring the synaptic weight of a superconducting optoelectronic loop neuron, a superconducting flux storage loop is inductively coupled to the synaptic current bias of the neuron. A standard flux memory cell is used to achieve a binary synapse, and loops capable of storing many flux quanta are used to enact multi-stable synapses. Circuits are designed to implement supervised learning wherein current pulses add or remove flux from the loop to strengthen or weaken the synaptic weight. Designs are presented for circuits with hundreds of intermediate synaptic weights between minimum and maximum strengths. Circuits for implementing unsupervised learning are modeled using two photons to strengthen and two photons to weaken the synaptic weight via Hebbian and anti-Hebbian learning rules, and techniques are proposed to control the learning rate. Implementation of short-term plasticity, homeostatic plasticity, and metaplasticity in loop neurons is discussed.