  The question of whether a hypercyclic operator $T$ acting on a Fr{\'e}chet
algebra $X$ admits or not an algebra of hypercyclic vectors (but 0) has been
addressed in the recent literature. In this paper we give new criteria and
characterizations in the context of convolution operators acting on $H(\mathbb
C)$ and backward shifts acting on a general Fr{\'e}chet sequence
algebra.Analogous questions arise for stronger properties like frequent
hypercyclicity. In this trend we give a sufficient condition for a weighted
backward shift to admit an upper frequently hypercyclic algebra and we find a
weighted backward shift acting on $c_0$ admitting a frequently hypercyclic
algebra for the coordinatewise product. The closed hypercyclic algebra problem
is also covered.
