Highly coherent and low-noise supercontinuum (SC) sources based on nonlinear spectral broadening of femtosecond pulses in all-normal dispersion (ANDi) fibers are attractive for many applications in ultrafast photonics. By simulating a real nonlinear pulse compression experiment, we numerically investigate the impact of shot noise and technical pump laser fluctuations on the quality and stability of single-cycle pulse generation and other multi-shot experiments based on the manipulation of the SC spectral phase. We find that for pump pulse durations of less than 600 fs, input relative intensity noise <1%, and correctly chosen fiber lengths, the initial fluctuations of the pump laser are at most amplified by a factor of three. We also show that the usual strong correlation between SC coherence and quality of the compressed pulses collapses in the presence of technical noise, and that in this situation the coherence is not a useful figure of merit to quantify pulse quality, noise amplification, or decoherence due to incoherent nonlinear dynamics. Our results highlight the very limited impact of technical pump laser noise on ANDi SC generation and are of practical relevance for many ultrafast photonics applications that require high-quality, low-noise SC sources.