As you are looking for some deep results, let me mention only two:
- the Kadec's-$1/4$ theorem that gives a stability bound under which perturbed exponential Riesz bases remain Riesz bases (M. I. Kadec, The exact value of the Paley-Wiener constant, Dokl. Akad. Nauk SSSR, 1964, see for instance R. Young, An introduction to non-harmonic Fourier series, p. 42)
- the theory called "finite rate of innovation" (Sampling Signals With Finite Rate of Innovation, IEEE Trans. Signal Proc., 2012, M. Vetterli et al.) which essentially generalizes Nyquist-Shannon theory to non band-limited signals, requiring "twice the number of degrees-of-freedom" per unit interval, like the two-points per period in the classical cases, for some parametric signals (finite number of spikes, piece-wise polynomials, etc.).
I like the genericity and the simple bounds for both examples.
You can probably find more ideas in Classical and approximate sampling theorems; studies in the $L^p(R)$ source and the uniform norm, J. Approx. Theor., 2005, P. L. Butzer et al. and Sampling Theory, a Renaissance: Compressive Sensing and Other Developments, 2015, G. E. Pfander editor. In the latter, you might like Chapter 9 - Sampling in Euclidean and non-Euclidean Domains: A Unified Approach (sphere, Hyperbolic Space), and Chapter 10 - A Sheaf-Theoretic Perspective on Sampling, where you get some topology.
There are all sorts of exiting developments in zero- or level-crossing, single-bit sampling, which I do not know enough to be more precise: round-off, jitter, truncation, saturation and aliasing errors are real problems.