Bypass capacitor causing op amp to oscillate

Question

I have a simple non-inverting op amp buffer, to buffer the input to my circuit. It works fine, but I wanted to add a bypass capacitor to the power rails to make sure that it is as low-noise as possible. I referred to this answer and decided to use one rail-to-rail capacitor because it was one less part. But, when I add the capacitor, it gets way more noisy.

^{simulate this circuit – Schematic created using CircuitLab}

When I zoom in on the noise, it actually looks like an oscillation at 35 kHz (see image below). The yellow trace is the output of my whole circuit. The blue trace is where I have re-created just the part shown in the diagram, with a single op-amp and capacitor on a breadboard with a constant voltage input. Both circuits are using the same power supply. When I take out the bypass capacitors, both lines return to being almost totally flat. What is going on here?

Answer 1

As the answer referenced by Bruce Abbott says, you need to use bypassing to ground, not between the rails. In your circuit, the load current flows through the impedance of your power supplies, providing a feedback path. ( complicated by the details of your input and load ground)