Recent nuclear magnetic resonance studies [A. Pustogow {\it et al.},
arXiv:1904.00047] have challenged the prevalent chiral triplet pairing scenario
proposed for Sr2RuO4. To provide guidance from microscopic theory as to
which other pair states might be compatible with the new data, we perform a
detailed theoretical study of spin-fluctuation mediated pairing for this
compound. We map out the phase diagram as a function of spin-orbit coupling,
interaction parameters, and band-structure properties over physically
reasonable ranges, comparing when possible with photoemission and inelastic
neutron scattering data information. We find that even-parity pseudospin
singlet solutions dominate large regions of the phase diagram, but in certain
regimes spin-orbit coupling favors a near-nodal odd-parity triplet
superconducting state, which is either helical or chiral depending on the
proximity of the γ band to the van Hove points. A surprising
near-degeneracy of the nodal s′- and dx2−y2-wave solutions leads
to the possibility of a near-nodal time-reversal symmetry broken
s′+idx2−y2 pair state. Predictions for the temperature dependence
of the Knight shift for fields in and out of plane are presented for all
states.Comment: 5 pages (3 figures) + supplementary informatio