I was wondering why the behavior is more "chaotic" at 0~1V when compared to 2V+ (which is when it starts to glow).
2 Answers
At very low voltages (and currents) the power produced is small, so the filament doesn't produce much heat. At these low temperatures the bulb tends to have significant interactions with the surroundings, especially drafts which will cool the bulb in an unpredictable fashion.
Another source of error may be the bulb contacts, which may be showing significant resistance change with very small changes in mechanical pressure/orientation. At higher currents, these effects become less significant.
You have two sources of error: error in the source voltage, and error in the current measurement. Since your graph is log-log, at the bottom left corner (small currents and voltages) any errors are greatly magnified; even a 50mA or 100mv error looks big. At the top right corner, however, the (probably of a similar magnitude) errors are swamped by the signal.