Multi-transverse-mode competition, coupling induced instabilities and multi-pulse or satellite pulse oscillations were investigated experimentally and theoretically in laser-diode-pumped Cr,Nd:YAG self-Q-switched microchip lasers under large pump beam diameter. The different transverse modes have great effects on the laser pulse temporal characteristics such as pulse profile, pulse width, instability of peak power and repetition rate jitter. Multi-transverse-mode, multi-pulse oscillation and periodical pulsation were observed by varying the pump beam diameter.The effect of transverse modes on the instability and multi-pulse oscillation were studied by modified coupled rate equations by taking into account the transverse-mode competition of inversion population under different pump conditions. The numerically simulated results are in good agreement with the experimental results. These results show that the multi-pulse oscillation and instability in the pulse train were attributed to different transverse mode coupling and competition. The peak power instabilities and pulse repetition rate jitter of the laser pulses due to transverse mode coupling were also investigated.