We will show that $\diamondsuit$ is consistent with the statement there is no minimal Souslin tree. This answers a question due to Baumgartner. We will also show that there is a Souslin tree $T$ whose restriction on any $C \subset \omega_1$ is rigid and forcing with $T$ makes $T$ a Kurepa tree. This answers a question due to Gunter Fuchs.