In joint work with G. Blekherman we proved that a truncated multisequence $y$ of degree $m$ has a representing measure in the Truncated Moment Problem if and only if its core variety $V(y)$ is nonempty, in which case $V(y)$ coincides with the union of the supports of all representing measures. In general, for a given numerical sequence $y$, it may be quite difficult to compute $V(y)$ or even to determine if it is nonempty. We illustrate some cases where we can compute $V(y)$ or can otherwise describe it concretely.