Softest, smoothest sounding capacitors you've tried?

Clarity CMR are smooth and transparent and add absolutely no coloration. Bypassed with 0.1uF Audyn TrueCopper they are absolutely perfect but I find the need to bypass less at small values.

All the fancy Mundorf’s add splash to the sound.

The Mundorf MKP are dark sounding caps with no splash.  Cheap so worth trying for the heck of it.

     Much of the, "vintage sound" you're referring to was generated by the carbon composition resistors used, back in the day.

      There are still a variety of Cornell-Dubilier, Black Cat capacitors available, that seemed a favorite, among many audio component designers as well, though: few in the higher uF values.

             ie: https://www.vivatubes.com/lot-of-4x-vintage-black-cat-black-beauty-axial-oil-capacitor-047uf-600v/

             and: https://www.ebay.com/sch/i.html?_from=R40&_trksid=p2047675.m570.l1313&_nkw=black+cat+capacitor&_sacat=0

Some very nice caps come in lower voltages, so choice depends if they are going to be used in an active circuit, close to tubes or not, or speaker xovers.

Check the avialiable space.

Agree with Erik, and i would add MR’s, both excellent, other option is Jupiter HT beeswax, and at lower cost the Obbligato Gold premium.

rodman99999 is correct also about carbon composition resistors.

I have been Duelunds for a while especially their newTinned Copper Foil 

he best capacitors,  but they are Big,

just Jupiter Copper  foil are possibly a bit smoother but not as much detail .

Of course, nothing is cast in stone, but in general, you are describing paper & oil caps