Capacitors for AMS1117 for 5V to 3.3V regulation
The 1117 has been cloned by a lot manufacturers. In general, if they only talk of tanatalum (as AMS does) they probably copied some old datasheet from somebody else. Look in others' datasheet (besides AMS). For example TI's LM1117 has more details:


  Output Capacitor:
  The output capacitor is critical in maintaining regulator stability, and must meet the required conditions for both
  minimum amount of capacitance and ESR (Equivalent Series Resistance). The minimum output capacitance
  required by the LM1117-N is 10µF, if a tantalum capacitor is used. Any increase of the output capacitance will
  merely improve the loop stability and transient response. The ESR of the output capacitor should range between
  0.3Ω - 22Ω. In the case of the adjustable regulator, when the CADJ is used, a larger output capacitance (22µf
  tantalum) is required.


And yes, output cap ESR is critical for the stability of LDOs, not just the capacitance. (Look in the LDO tag for some recent on-site examples; this one in particular has scope traces.) The NCP1117 datasheet is most helpful in this regard:


  Frequency compensation for the regulator is provided by
  capacitor Cout and its use is mandatory to ensure output
  stability. A minimum capacitance value of 4.7µF with an
  equivalent series resistance (ESR) that is within the limits of
  33 mΩ (typ) to 2.2Ω  is required. See Figures 12 and 13. The
  capacitor type can be ceramic, tantalum, or aluminum
  electrolytic as long as it meets the minimum capacitance
  value and ESR limits over the circuit’s entire operating
  temperature range. Higher values of output capacitance can
  be used to enhance loop stability and transient response with
  the additional benefit of reducing output noise.


[emphasis mine]. That datasheet has nice graphs that show the interdependence of these two output cap parameters (ESR and capacitance):



Now the AMS1117 datasheet says nothing about ESR, so you can believe theirs is godly and won't oscillate regardless of that, but I rather doubt that's actually the case. This is all they say, by the way:


  The circuit design used in the AMS1117 series requires the use of
  an output capacitor as part of the device frequency compensation.
  The addition of 22µF solid tantalum on the output will ensure
  stability for all operating conditions.
  When the adjustment terminal is bypassed with a capacitor to
  improve the ripple rejection, the requirement for an output
  capacitor increases. The value of 22µF tantalum covers all cases of
  bypassing the adjustment terminal. Without bypassing the
  adjustment terminal smaller capacitors can be used with equally
  good results.
  To further improve stability and transient response of these
  devices larger values of output capacitor can be used. 




Now regarding


  Actually "Getting started with STM32F030xx" recommends using 4.7uF capacitor between VCC and GND (in addition to some 100nF) which is not that much smaller than the recommended one.


That's for the decoupling of the STM32F itself


  In addition, each power supply pair should be decoupled with 100 nF filtering ceramic
  capacitor and a chemical capacitor of about 4.7 μF connected between the supply pins of
  the STM32F030 device. These capacitors need to be placed as close as possible to, or
  below, the appropriate pins on the underside of the PCB.


Between the MCU and the regulator you may have non-trivial inductance. The pros simulate the board in a rather expensive EM field simulator to figure out how much exactly. If you can't afford that, you have to go with the rules of thumb from the IC manufacturers' datasheets.

Note that this EM simulation is only about the effect of the board, it does not include things like stability of voltage regs. And if you put the two chips and their caps really close to each other, the lack of trace inductance/resistance between them may actually be a problem for the stability of the LDO: the capacitance it sees grows (sees caps in parallel), but the ESR goes down (resistors in parallel). That moves you to the left in graph in Fig 12 above.
That's why nothing is ultimately the substitute for actual prototyping.