I have a pcb with the following circuit.
The inputs (left side) are small DC voltages, within ~100mV of ground. They won't be DC in the final application, but for debugging that's what they are. No gain resistors are used. There are decoupling caps on the +/- 15V power supply (0.1uF ceramic 0402 close to pins, 10uF tantalum 1206 next, then 1mF electrolytic ~3cm from pins). It's a lab power supply, very clean.
The output signal is a very clean oscillation ~1V peak to peak, ~6 MHz. Shown below are the inputs (traces 3 and 4, purple and pink respectively), and the output in green.
When I add a 10nF cap in parallel with the 1kOhm output resistor, R5, I get the trace below, with a ~2MHz frequency but similar amplitude.
I added a 10uF ceramic capacitor in parallel with R5, and the oscillations were either small enough that I couldn't see them on the scope any more or they were gone entirely. I may be able leave the output cap in my final circuit, but I would be much more comfortable with doing so if I understood where this problem was coming from.