Power regulator circuitry for programmable memory elements on programmable logic device integrated circuits is provided. The programmable memory elements may each include a storage element formed from cross-coupled inverters and an address transistor. Address drivers may be used to supply address signals to the address transistors. The power regulator circuitry may include an address power supply circuit that produces a time-varying address power supply voltage to the address drivers and storage element power supply circuits that provide time-varying storage element power supply voltages to the cross-coupled inverters in the storage elements. Unity gain buffers may be used to distribute a reference voltage from a bandgap voltage reference to the power supply circuits. The power supply circuits may use voltage dividers and p-channel metal-oxide-semiconductor control transistors.