  At a temperature of roughly 1\,K, \ce{Sr2RuO4} undergoes a transition from a
normal Fermi liquid to a superconducting phase. Even while the former is
relatively simple and well understood, the superconducting state is not even
after 25 years of study. More recently it has been found that critical
temperatures can be enhanced by application of uniaxial strain, up to a
critical strain, after which it falls off. In this work, we take an
`instability' approach and seek for divergences in susceptibilities. This
provides an unbiased way to distinguish tendencies to competing ground states.
We show that in the unstrained compound the singlet and triplet instabilities
of the normal Fermi liquid phase are closely spaced. Under uniaxial strain
electrons residing on all orbitals contributing to the Fermiology become more
coherent while the electrons of Ru-$d_{xy}$ character become heavier and
electrons of Ru-$d_{xz,yz}$ characters become lighter. In the process,
Im\,$\chi(\mathbf{q},\omega)$ increases rapidly around the incommensurate
vector $\mathbf{q}{=}(0.3,0.3,0)2\pi/a$ while it gets suppressed at all other
commensurate vectors, in particular at $q{=}0$, which is essential for
spin-triplet superconductivity. Thus the triplet superconducting instability
remains the lagging instability of the system and the singlet instability
enhances under strain, leading to a large energy-scale separation between these
competing instabilities. At large strain an instability to a spin density wave
overtakes the superconducting one. The analysis relies on a high-fidelity,
\emph{ab initio} description of the one-particle properties and two-particle
susceptibilities, based on the Quasiparticle Self-Consistent \emph{GW}
approximation augmented by Dynamical Mean Field theory. This approach is
described and its high fidelity confirmed by comparing to observed one- and
two-particle properties.
