Thanks CompoundCarl : I lowered the res to 1, and it put support under them. I could complain that the support is too small and sparse now, so I still now need to manually go in and add more support just so the tiny support towers won't try and tip over during print, but at least that's better.
Unfortunately, this doesn't fix my previous issues/examples, where spikes are hanging down, since they terminate at an infinitely small point, they get clipped out of the "s3d support filter" it seems: The feature is still smaller than 1mm (infinitely small at the vertex) so no "automatic" support for you, regardless of what you set the support resolution to.
Makerbot Replicator 2x / Sailfish on Mightyboard
Custom C-Bot / RepRap Firmware on RADDS
Custom Tevo Little Monster / RepRap Firmware on Duet