Depends on what the supply is, and more crucially, what the loop
response, output ripple and step response has to be.
The issues are input and output operating voltage (above about 10V,
there aren't many MLCCs of high capacitance that readily available
anyway), output capacitance requirements, output ESR, input ripple and
surge, and the controller you are using.
The tradeoffs are:
Electrolytics have high capacity, reasonable ripple RMS ratings and
relatively high ESR (although the panasonic special polymer series are
quite low). Good choice especially at the input of a power supply for
input voltages above about 8VDC. The special polymer series are quite
effective on outputs too.
Tantalums have low esr (in their latest incarnation - see the AVX TPS
series for instance), but have a habit of spectacularly failing for
high surge currents into the device, even well below their rated
voltage. For this reason, they are not usually recommended at the input
of a regulator. They are fine at the output, where there are large
discharge currents, not [usually] high surge currents.
MLCC have the lowest esr available (below 2milliohm on certain devices)
at up to 100uF for 6.3V, but this is not a panacea. Low output ESR can
play havoc with loop compensation.
My usual approach is to use some tantalums with the bulk of the
capacitance at the output with a couple of MLCC parts (about 1/5 or
less total Cout) to lower the esr for small signal ripple performance.
At the input, it depends on Vin - I'll use tantalums, but overrate them
by at least a factor of 2 for both rated voltage and rated RMS ripple
current for safety. I would use electrolytics, but they just tend to be
too big for my applications. If the input voltage is low enough, I'll
use MLCCs.
For Cout, using a mixture of tants and MLCCs gives me an acceptable
Cout/ESRout and superior ripple performance. It won't work for
everyone, obviously, and much depends on Vout (and range of load
currents too).
MLCCs aren't that expensive, incidentally, especially given their life
and temperature stability (Winfield Hill posts here, a co-author of
'The Art of Electronics', which has an amusing table of the various
characteristics of capacitor types - look under temperature stability)
There really is no simple answer - every supply has to be designed with
it's specific requirements in mind.
Cheers
PeteS