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Continuously pumped Yb-doped double-clad fiber lasers exhibit two kinds of instabilities, known as self-sustained pulsing and self-mode locking (SML). We study numerically the role of self-phase modulation (SPM) on the onset of these stabilities by using a spatiotemporal model that takes into account fully the length dependence of both the optical gain and spontaneous emission, while also including the nonlinear effects within the fiber. Self-sustained pulsations originate from relaxation oscillations. Although they die out quickly in the absence of SPM, our results show that SPM restores these oscillations when the laser power exceeds a certain value. SML has its origin in the spontaneous-emission noise acting as a seed. The laser cavity amplifies this noise the most at frequencies associated with its longitudinal modes, resulting in periodic power variations with an amplitude <; 1% (without SPM). We find that SPM plays again a critical role and increases their amplitude to above the 10% level. We also show that the SPM-induced instabilities can be suppressed by inserting a passive fiber of suitable length inside the laser cavity, which is in agreement with recent experimental work.