  We study Banach spaces whose group of isometries acts micro-transitively on
the unit sphere. We introduce a weaker property, which one-complemented
subspaces inherit, that we call uniform micro-semitransitivity. We prove a
number of results about both micro-transitive and uniformly
micro-semitransitive spaces, including that they are uniformly convex and
uniformly smooth, and that they form a self-dual class. To this end, we relate
the fact that the group of isometries acts micro-transitively with a property
of operators called the pointwise Bishop-Phelps-Bollob\'as property and use
some known results on it. Besides, we show that if there is a non-Hilbertian
non-separable Banach space with uniform micro-semitransitive (or
micro-transitive) norm, then there is a non-Hilbertian separable one. Finally,
we show that an $L_p(\mu)$ space is micro-transitive or uniformly
micro-semitransitive only when $p=2$.
