Can we measure an electromagnetic field? As far as I can check, the Aharonov-Bohm effect is not -- contrary to what is claimed in the historical paper -- a demonstration that the vector potential $A$ has an intrinsic existence in quantum mechanics, neither that the magnetic field $B$ has an intrinsic existence in classical mechanics, but rather that the magnetic flux is the relevant, measurable quantity. Indeed, the phase difference is either $\int A \cdot d\ell$ or $\int B \cdot dS$ (in obvious notations with surface $dS$ and line $d\ell$ infinitesimal elements), and these two quantities are always equal, thanks to the theorem by Stokes. 
Then, one could argue that classically one can measure the flux as defined above, whereas in QM one has only the Wilson loop $\exp \left[\mathbf{i} \int A \cdot d\ell / \Phi_{0}\right]$, with $\Phi_{0}$ the flux quantum, as a relevant, measurable quantity.
I just elaborated above about the Aharonov-Bohm effect since it was the first time I realised the possible experimental difference between measuring a flux and a field. Let me now ask my question.
My question is pretty simple I believe: Does anyone know a classical experiment measuring the magnetic field $B$ and not the flux $\int B \cdot dS$ ?
As long as we use circuits the answer is obvious, since one can only have an access to the flux and the voltage drop $\int E \cdot d\ell$ ($E$ the electric field), but do you know other experiment(s)?
As an extra, I believe the same problem arises with force: *Does anyone know a classical experiment probing the force field $F$, and not the work $\int F \cdot d\ell$ ? *
NB: The question does not rely on how smart you can be to find a protocol that gives you an approximate local value of the field, say $\int F \left(x \right) \cdot d\ell \rightarrow F\left(x_{0} \right)$ with $x_{0}$ a (space-time) position. Instead, one wants to know whether or not a classical measurement can a priori gives a local value of the field, without a posteriori protocol. Say differently, one wants to know how to interpret a classical measure: is it (always ?) an integral or can it -- under specific conditions -- give a local value ?
Post-Scriptum: As an aside, I would like to know if there is some people already pointing this question out (wether it's the flux or the local magnetic field or vector potential that one can measure), and if it exists some reference about that in the literature.
 A: The simplest device to measure magnetic fields (and not integrated magnetic flux) is an electron. The spin-up/-down energy splitting in a magnetic field is  $g_S\mu_B B$ where $g_S \mu_B \sim 2.8~\mbox{MHz/G}$. This can be measured in a beam machine. The next simplest is the Zeeman splitting of atomic energy levels.  The latter forms the basis of some of the most sensitive magnetometers, for example John Kitching's microfabricated gas cell magnetometers (external link to a pdf describing the device)  at NIST. 
Building upon Ben Crowell's comment, a balance scale will measure the gravitational force on an object while keeping $\Delta l$ arbitrarily small (the more accurate your measurement of the scale's tilt, the less the masses move during the measurement).
A: Measure the Lorentz force acting on a moving point charge, e.g. electron in a Penning trap or cyclotron resonance.
A: A compass is a classical magnetic field measurement.  It's nontrivial to get the magnitude from a compass but the direction is certainly easy.  But if you read Feyman you'll see that it is impossible to explain magnetism classically.  Our models of electrons in teeny Amperian loops is bogus.  In fact the famous Bohr–van Leeuwen theorem shows that classical magnetic fields are impossible.  So magnetism is fundamentally a quantum mechanical phenomenon.
And here's a nit to pick with some of the answers.  Almost no measurement is direct.  A bathroom scale does NOT measure force.  If it has a spring, it measures displacement and converts that to force via Hooke's law. Thus you're assuming you know the spring constant and that the spring is kept in it's linear regime.  If it's a strain gage device it doesn't even directly measure strain directly.  It measure tiny changes in resistance using a Wheatstone bridge and converts these to strain and thence to "weight." Almost every measurement you can think of is actually an inference of what you want from what you can really measure.
A: If the magnetic field from the solenoid is strong enough you might be able to place a physical magnet on a sensitive force meter to provide a reading from which the strength of the magnetic field could be calculated. I have never heard of it being done except in relatively simple situations. The force between bar magnets is the calculation that would be necessary to carry out. The setup would entail treating the solenoid as one of the two magnets. This may or may not be possible. I have no experience doing this. I am merely pontificating like a dilettante. 
This was meant more as a comment than an answer, but considering the large number of comments I desired to keep it from getting lost in the discussion.

Say differently, one wants to know how to interpret a classical measure: is it (always ?) an integral or can it -- under specific conditions -- give a local value ?

Unless you are willing to infer a local value from the trajectory of an electron as Jason mentioned, or from the same bending of electrons in the hall effect sensors, then there is always going to be some area involved. However if area/flux is the issue to avoid then you could always try the Gilbert model, which Griffiths recommends against using for any quantitative work.  
A: A small rotating loop can measure B,  small rotating short dipole can measure E.  A to my knowledge, has never been measured. Remarkably the secondary winding of an iron cored transformer operates in an area devoid of B field but there is significant A.  The AB effect is not proved by the the famous Tonomura experiment.  There are peer reviewed articles critical of this experiment.  One of the papers is titled:Is the Static Magnetic Vector Potential Screened by a Thick Superconducting Shield
