  The superconductor Sr2RuO4 has been the subject of enormous interest over
more than two decades, but until now the form of its order parameter has not
been determined. Since groundbreaking NMR experiments revealed that the pairs
are of dominant spin-singlet character, attention has focused on time-reversal
symmetry breaking linear combinations of $s$-, $d$- and $g$-wave
one-dimensional (1D) irreducible representations. However, a state of the form
$d_{xz}+id_{yz}$ has also been proposed. We present a systematic study of the
stability of various superconducting candidate states, assuming that pairing is
driven by the fluctuation exchange mechanism, including a realistic
three-dimensional Fermi surface, full treatment of both local and non-local
spin-orbit couplings, and a wide range of interaction parameters $U,J,U',J'$.
The leading superconducting instabilities are found to exhibit nodal
even-parity $A_{1g} (s')$ or $B_{1g} (d_{x^2-y^2})$ symmetries, similar to the
findings in two-dimensional models without longer-range Coulomb interaction
which tends to favor $d_{xy}$ over $d_{x^2-y^2}$. Within the so-called Hund's
coupling mean-field pairing scenario, the $E_g (d_{xz}/d_{yz})$ solution can be
stabilized for large $J$ and specific forms of the spin-orbit coupling, but for
all cases studied here the eigenvalues of other superconducting solutions are
significantly larger when the full fluctuation exchange vertex is included in
the pairing kernel. Additionally, we compute the spin susceptibility in
relevant superconducting candidate phases and compare to recent neutron
scattering and NMR Knight shift measurements.
