We consider the problem of optimal experimental design (OED) in the context of PDE-constrained inverse problems. We start with an iterative (greedy) sensor placement strategy based on a stability analysis of a deterministic inverse problem. In this approach, the measurement functionals are chosen to improve an observability coefficient related to the stability of the estimation problem. We then connect this work to Bayesian inversion, particularly to more standard Bayesian OED techniques based on A-, D, and E-optimality criteria. We then relate the sensor selection to the ability of detecting both modelled and un-modelled phenomena and we reflect on the implications on OED and on data assimilation, in general, for problems involving modelling errors and biases.