The halting problem is undecidable, i.e. $\not \exists$ $M$ Turing machine s.t. for every $(M_0,w_0)$ input where $M$ is the description of a Turing machine and $w_0$ is an input word, the output of $M$ is "$1$" if $M_0$ will halt on $w_0$ and "$0$" if it will not.
From this statement only, it does not follow that there must be an ($M_0,w_0$) pair for which it is impossible to predict if $M_0(w_0)$ halts or not. It only means there is no general way to decide this question for all ($M_0,w_0$) pairs.
So my question is the following: Does there exist a ($M_0,w_0$) pair for which it is known that it is impossible to predict whether $M_0(w_0$) halts or not? If yes, is there a known example or construction for an ($M_0,w_0$) pair such that?