  The unconventional superconductivity in Sr$_2$RuO$_4$ continues to attract
considerable interest. While many measurements can be interpreted on the basis
of chiral $p$-wave pairing with intriguing topological character, a number of
exceptions hinder an unambiguous verification of such pairing. The pairing
mechanism also remains under debate. In this paper, with effects of the sizable
spin-orbit coupling accounted for, we reexamine the superconducting
instabilities in Sr$_2$RuO$_4$ through systematic microscopic analysis within
random phase approximation. Our calculations show that the odd-parity $p$-wave
pairing is favored in the regime of extremely weak interactions, but that
highly anisotropic even-parity pairings become most leading over a broad range
of stronger interactions. These results could shed light on the nature of the
enigmatic superconductivity in Sr$_2$RuO$_4$.
