Why Vp/Vs and not Vs/Vp?
After quite a bit of conversation on Twitter, I think it's arbitrary. Tradition is probably the prevailing reason.

It would be interesting to go back through the literature to see who first used it. My hunch is that it goes back to the Zoeppritz equations (Zoeppritz 1919), which feature both $V_\text{S}/V_\text{P}$ and $V_\text{P}/V_\text{S}$. I bet Aki &amp; Richards (1980) also had something to do with its propagation, at least in the exploration geophysics community. 

Thinking about it, I think there are some good reasons to use $V_\text{S}/V_\text{P}$ instead of $V_\text{P}/V_\text{S}$.

It's well-behaved

As the OP said, $V_\text{S}/V_\text{P}$ behaves nicely, being constrained in the range [0,1) for rocks and fluids (and weird things like bitumen) — compared to $V_\text{P}/V_\text{S}$, which varies in the range 1 to infinity/undefined. I suppose you could call that 'neat'...

It's a good gas indicator

If you have a background of brine (say) and want to see gas anomalies, then my insufficient research suggests that $V_\text{S}/V_\text{P}$ is actually a better discriminator:



It's in the Aki–Richards equation

This is the formulation from Avseth et al. (2006):

$$ R(\theta) = \frac{1}{2} \frac{\Delta \rho}{\rho} - 2 \left( \! \frac{V_\mathrm{S}}{V_\mathrm{P1}} \! \right)^2 \frac{\Delta \rho}{\rho} \sin^2 \theta + \frac{1}{2} \frac{\Delta V^2_\mathrm{P}}{V^2_\mathrm{P}} \frac{1}{\cos^2 \theta_\mathrm{avg}} - 4 \left( \! \frac{V_\mathrm{S}}{V_\mathrm{P1}} \! \right)^2 \frac{\Delta V^2_\mathrm{S}}{V^2_\mathrm{S}} \sin^2 \theta $$

Refer to the book or SubSurfWiki for the symbol definitions etc. 

Hmm, none of these are reasons why $V_\text{P}/V_\text{S}$ prevails. This is starting to look like the $\pi$ vs $\tau$ debate...

References


Aki, K, and PG Richards (1980). Quantitative Seismology: Theory and Methods. WH Freeman and Co.
Avseth, P, T Mukerji, G Mavko (2006). Quantitative Seismic Interpretation. Cambridge University Press.
Castagna, J, and H Swan (1997). Principles of AVO crossplotting. The Leading Edge, Society of Exploration Geophysicists, 17, 337–342. 
Smith, T, et al (2003). Gassmann fluid substitutions; a tutorial. Geophysics (April 2003), 68(2):430-440. DOI 10.1190/1.1567211. 
Zoeppritz, KB (1919). Erdbebenwellen VII. VIIb. Über Reflexion und Durchgang seismischer Wellen durch Unstetigkeitsflächen. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-physikalische Klasse, 66-84.