Planets classification by density
In our Solar system, the density of major planets varies from 0.7 g/cm3 (Saturn) to 5.5 g/cm3 (Earth).
For exoplanets, the density varies from very low (0.03 g/cm3 for Kepler-51x) to very high (77.7 g/cm3 for Kepler-131c, data according to NASA Exoplanet Archive).
We know that bodies with low density are composed of gas and others are rocky planets. Wikipedia provides some approximations about it.
Is there any detailed classification of planetary bodies by density? A classification that would say, for example, that the density below 1.5 g/cm3 corresponds to a gas planet; between 1.5 and 3 are icy; between 3 and 8 are rocky; above 8 g/cm3 are even more rocky etc...
Would such a classification make sense?
If it exists, which are the numeric bounds for the different types of bodies?
There is no 1:1 mapping between density and composition/structure. You have to look at detailed planetary models. For example, some hot Jupiters are extremely dense ($\geq 10$ g/cm$^3$) but they are undoubtedly gas giants. The origins of this diversity are the source of much speculation and theory, but are certainly within the realms of known physics.
An example of the difficulties can be gleaned from this plot from Lissauer et al. (2014) that shows the mass-radius plane for small planets together with the loci of models for various compositions. Lines of constant density are drawn on the plot too, but notice that a planet of a given composition can have a range of densities depending on its mass.
The situation appears even more strange with giant planets. Generally speaking, more massive giant planets are more dense, which is as expected from basic theory of degenerate gases, but objects of a given mass can have almost an order of magnitude spread in density. Below is a plot I made from exoplanets.org. Almost certainly, the change in the nature of the relationship at around 0.1 Jupiter masses marks the transition from gas giants to ice giants and other rocky-type planets, but notice that gas giants can be just as dense as smaller planets. i.e. A classification based on just density wouldn't work, but mass and density gives more of a clue.