Detailed polarization properties have been examined in laser-diode-pumped (LD-pumped) micro-grained ceramic Nd:YAG lasers in different microchip cavity configurations. Stable linearly-polarized single-frequency oscillations, whose polarization direction coincide with that of an LD pump light, were observed in an external cavity scheme. While, in the case of a thin-slice laser scheme with coated reflective ends, elliptically-polarized single-frequency operations took place in the low pump-power regime and dynamic instabilities appeared, featuring self-induced antiphase modulations among counter-rotating circularly-polarized components having slightly different lasing frequencies, with increasing the pump power. PACS: 42.55. Xi,42.55.Rz,42.25.Ja,42.60.Mi