Measuring extra-dimensions
In addition to what dmckee said, another hint at ("large") extra dimensions would be the detection of Kaluza-Klein particles at the LHC for example.
Kaluza-Klein particles are in principle nothing but the known standard model particles which can propagate into the extra dimensions if these are large enough. It can be shown that the angular momentum in these extra dimensions is quantized. This leads to the effect that particles propagating into the extra dimensions would be observed as heavier versions of the known standard model particles due to the additional momentum in the otherwise not directly visible dimensions. The energy (or mass squared) spectrum of the corresponding expected particle tower would have a step size proportional to 1/r (where r is the radius of the extra dimension).
As Prof. Strassler explains here, to determine the shape and extent of such large extra dimensions it would be necessary to measure the whole mass spectrum using more than one KK particle.
Up to now no KK particles have shown up at the LHC so far (which was run only at 7TeV and now continues at 8 TeV). But note that even if there could be such large extra dimensions leaving hints at themselves at the "LHC scale" (up to 14 TeV), this does not have to be the case for ST to work; the "large" extra dimensions are only a feature of certain (phenomenologica) models ...
Make some assumptions about the physics associated with the dimensions in questions (say electric field strength goes by $r^{-(n-1)}$ over distances in which $n$ dimensions are significant).
Make predictions on that basis
Compare to experiment
Many predictions can be made and tested in the realms of high energy particle physics, but so far all are null.
The standard way to measure compactified dimensions is to test some inverse-square law (e.g. Newton's, electromagnetic, diffusion) at the scale and see if it breaks down and starts approaching some other (higher power) inverse-power law.
In fact, the inverse-square law has only been verified down to a scale of 0.1mm -- here's a recent experimental paper doing this: [1].
(Yes, you can measure time in metres, by multiplying by the speed of light. This is where "lightseconds" and other such measurements of distance come from. An example motivation for treating this as the unit of the time dimension is from the Minkowski metric, $ds^2=c^2dt^2-dx^2-dy^2-dz^2$, where $ct$ is a dimension analogous to the spatial ones.)