measurements - How to measure Entropy?

While $$\delta Q=T\ dS$$ allows for obtaining the entropy change of an isolated system in equilibrium by measuring the heat exchange with the environment, I was wondering whether there are ways to obtain an entropy field distribution, potentially also in non-equilibrium systems?

Answer

Regarding your new formulated question "my question boils down to whether a device similar to a voltmeter or thermometer exists for entropy":

Well in that case then the answer is no, you can only measure entropy by studying the change of its dependencies on extensive/intensive variables that influence it (as they are different in different systems). To make more contrast, take the example of liquid crystals, where one has to define an appropriate order parameter to study and to be able to quantify the entropy change of the system. (e.g. nematic order parameter, density, etc.)

EDIT: An edit to further elaborate on the matter at hand, since from the comments it is clear that some folks may still not be convinced.

Quantities like entropy or chemical potential do not come about as directly accessible in a lab. What we call "measurable" typically comes with a mechanical response like pressure, bulk quantities or thermal ones like temperature and heat flow (i.e. studying $T$ changes in coupled systems). Hence we call $T$, $P$, $V$, $\Delta H_{latent}$ etc. measurable thermodynamic variables. Additionally there are also response functions that are measurable, as they correspond to a change in a system parameter in response to a perturbation, e.g. the expansion coefficient $\alpha_{p} \propto \delta V / \delta T$ at constant $P$, or heat capacities $C_V$, $C_P$, of course this is barely the full list (considering also combined response functions).

Now having mentioned a handful of measurable variables, next step is to see how these make the experimentally inaccessible quantities measurble. This is done using the famous Maxwell relations, an example:

Starting from an expression containing non-measurable quantities: (Using the expression of the expansion coefficient) $$\begin{align} \left(\frac{\delta S}{\delta p}\right)_{N,T} &= - \left(\frac{\delta V}{\delta T}\right)_{N,P} \\ \left(\frac{\delta S}{\delta p}\right)_{N,T} &= - V\alpha_{P} \end{align}$$
Which simply gives us the entropy dependance on pressure (at constant $T$) is related to thermal expansion of a system. One can keep using Maxwell relations to further simplify the expressions and ultimately reach relations only containing measurable ones.