What I suggested was:
- For most cases, power decoupling capacitors for the regulators should be ~100nF "high-K ceramic capacitors" + 47uF~100uF "electrolytic capacitors".
- For opamps, 100nF high-K ceramic should be fine, but you should consult with datasheets.
- Usually, you don't need to use tantalum capacitors for this purpose unless specified.
- Don't use film capacitors for power decoupling.

79XXs are less stable compared to 78XXs, and tend to become unstable depending on the load capacitance.
One should consult with the datasheet of each chip in order to know the proper capacitors values.
But also, you may need to tweak the capacitor value when necessary. Above recipe works most of the case.