Loop quantum cosmology (LQC), mainly due to Bojowald, is not the cosmological sector of loop quantum gravity (LQG). Rather, LQC consists of a truncation of the phase space of classical general relativity to spatially homogeneous situations which is then quantized by the methods of LQG. Thus, LQC is a quantum-mechanical toy model (finite number of degrees of freedom) for LQG (a genuine QFT with an infinite number of degrees of freedom) which provides important consistency checks. However, it is a non-trivial question whether the predictions of LQC are robust after switching on the inhomogeneous fluctuations present in full LQG. Two of the most spectacular findings of LQC are that: (1) the inverse scale factor is bounded from above on zero-volume eigenstates which hints at the avoidance of the local curvature singularity and (2) the quantum Einstein equations are non-singular which hints at the avoidance of the global initial singularity. This rests on (1) a key technique developed for LQG and (2) the fact that there are no inhomogeneous excitations. We display the result of a calculation for LQG which proves that the (analogon of the) inverse scale factor, while densely defined, is not bounded from above on zero-volume eigenstates. Thus, in full LQG, if curvature singularity avoidance is realized, then not in this simple way. In fact, it turns out that the boundedness of the inverse scale factor is neither necessary nor sufficient for the curvature singularity avoidance and that non-singular evolution equations are neither necessary nor sufficient for initial singularity avoidance because none of these criteria are formulated in terms of observable quantities. After outlining what would be required, we present the results of a calculation for LQG which could be a first indication that our criteria at least for curvature singularity avoidance are satisfied in LQG.