The basic criteria of phase margin, gain margin and loop stability is covered in any number of basic texts and is covered in detail in specific control loop theory texts.
Basically a circuit would be considered an oscillator (unstable) if the phase falls below 0 degrees while the gain is still positive. Phase margin and gain margin are figures of merit for how much margin exists in a circuit form oscillation. The phase margin is the amount of phase in degrees where the gain reaches 0 dB. gain margin is the gain attenuation in dB when the phase falls to 0 degrees. The general figures of merit or phase margin of at least 45 degrees and gain margin of -12 dB. 60 degrees of phase margin is better. I will sometimes go with less if the phase plot is fairly flat in the region of the crossover frequency. So for that criteria, plot 1 is far superior to plot 2.