Note that where you live is pretty important to this question. Since you are in Long Beach, where it never gets particularly cold, the 120V outlet is a viable option. In the land of Winter, 120V would most likely degrade into 0mph of charge.
Given you already have a subpanel right there, I'd probably throw a breaker and outlet on there anyway. Its what, maybe $15 for a breaker, and another $10 for wire/box. Size of breaker and type of plug/wire would be dictated by the amount of power available at the panel, and how cautious you are. Its pretty easy to relatively safely install a 20 amp/240 volt circuit. When you step into the 40-50 amp range and higher is where people start causing fires because they didn't use the right wire, didn't properly tighten the lugs, and so on.
To put 120V charging in perspective, I've been charging at home for a week(well, a couple nights anyway). I wasn't expecting much, and I pretty much got what I expected. First I tried a heavy(12gauge) extension cord(25', I think) into one of the overhead garage door plugs. Got 2mph at 8 of 8 amps, with a whopping 105V. I know for sure my garage is fed by a 14 gauge wire and 15 amp breaker.
Tried another 120V outlet that's fed by a 30 amp pool subpanel(not one the garage gets power from) in the garage, again with the same extension. Same 2mph.
Then I tried another heavy extension cord, this one 50-75' into a different outlet on the front of my home that I know is fed by 10 gauge wire and a 20 amp breaker. Again, 2mph, but I didn't note the voltage. This circuit had no other load.
Finally I returned to the garage, and plugged directly into the wall outlet, cutting out the extension cord and maybe 30' of 14 gauge. Now at least I got 3mph, occasionally 4mph, again at 105v, but it wavered between 9 and 12 of 12 amps.