Value of the capacitor is not critical, for 500mA load it is enough 470uF and 0.1uF on both sides. If you are dealing with low signal (1mV) then you consider the noise. By theory tantalum capacitor shouldbe the best in ptactice they ar not.
To find out you have a good or not capacitor is look at Vcc on osciloskope. Not long time ago I had a problem with noisy capacitors, I was trying about 20 of them and find only one electrolic working ok ( I have 10 of them fron one supplyier and every one of it is ok )
And the noice was not generated by power supply but in op amp woltage divider.
SoI threw away op amp and replaced it by single transistor - noise 100 lower.