The correct answer is $\boxed{\text{A. None follows}}$.
The first statement, “Some towels are brushes,” can be expressed in propositional logic as $\exists x(T(x) \land B(x))$. The second statement, “No brush is soap,” can be expressed as $\forall x(B(x) \rightarrow \neg S(x))$. The third statement, “All soaps are rats,” can be expressed as $\forall x(S(x) \rightarrow R(x))$.
The first conclusion, “Some rats are brushes,” can be expressed as $\exists x(R(x) \land B(x))$. This conclusion does not follow from the premises, because the premises do not tell us anything about whether or not there are any rats.
The second conclusion, “No rat is brush,” can be expressed as $\forall x(R(x) \rightarrow \neg B(x))$. This conclusion also does not follow from the premises, because the premises do not tell us anything about whether or not there are any rats.
The third conclusion, “Some towels are soaps,” can be expressed as $\exists x(T(x) \land S(x))$. This conclusion also does not follow from the premises, because the premises do not tell us anything about whether or not there are any towels.
Therefore, none of the conclusions follow from the premises.