The author proves several results of the following flavour: given an important ring-theoretical property \(\Theta\), he describes (both structurally and in the language of identities) all semigroup varieties \(V\) such that for each (or for each finite, or for each locally finite) semigroup \(S\in V\), the semigroup algebra \(FS\) over a field \(F\) obeys the property \(\Theta\). The properties involved are: ``to be right Artinian as a ring'' (here \(S\) is assumed to be finite), ``to satisfy a polynomial identity'', ``to be von Neumann regular'', ``to be local'', ``to be semilocal'' (in the last 3 cases \(S\) is supposed to be locally finite).