I might be trying to upsample some stereo soundtracks from a series of shorts clips and was wondering which one of these two upsampling methods would you use and why. Also, are there any point-and-click solutions for either of them that are either freeware or cheap?

1) The upsampling method is relative. U have to use what U like best. You need to conduct our own testing and see what sounds best for you. There´s no objective choice between them.
2) The cheapest (free) and fastest upsampling i know can be done thru GraphEdit and filters like ffdshow or matrixmixer. I think foobar2000 have matrix mixing capabilities, not sure, i don´t use that program myself.

I know upsampling quality in itself is subjective, but those these two methods are based on theoretical procedures and are therefore objective and open to debate. Since there are many members here that really know their stuff I'd like to hear their opinions on the matter so that I have a good starting point from which to make a good, informed decision.

I think what you mean is called upmixing.
Apart from being subjective to taste it also depends on the source what work best.

I'll give it a shot.
- DPLII more to the rear channels but sometimes unnatural placement.
- NEO:6 more natural (ambience) but sometimes a bit dull surround experience.

It is upmixing, an one also has to consider that both technologies allow some user-tweaking of parameters. For DPL II, 'Center Width', 'Panorama' and 'Dimension' are all user-configurable, and produce quite audible differences. It would be hard to set up a perfectly fair comparison between the two upmixing DSPs, other than comparing their 'default' settings..

I recomend you to try allso Pro_optimizers foo_dsp_fsurround upmixing, as tebasunda said its has most exact channel separation i with foobar you can easily convert any source into wav using this dsp and get 2ch->5.1ch upmix if you need lfe there is chanel mixerplugin with you can put aftewards.
I hope there will be more updates form pro_optimizer

I agree with krabapple and poisas. I tend to preffer NEO:6 because it sounds more natural to me and generally (in music mode) preserves the original stereo image. If your effect has the full controls that poisas mentioned (some has only the mode switch - movie and music), you can achieve similar results with DPLII if you switch off the panorama and don't use high value for dimension.
NEO:6 has only one parameter in music mode. CGAIN. It controls the amount of subtraction of derived center channel from L and R channels. It ranges from 0 (no subtraction) to 0,5 (-6 dB). Default value is 0,2.