  The large-scale behavior of two-dimensional critical percolation is expected
to be described by a conformal field theory (CFT). Moreover, this putative
percolation CFT is believed to be of the logarithmic type, exhibiting
logarithmic corrections to the most commonly encountered and studied behavior
of CFT correlations.
  It was recently shown by the first author [Cam24] that critical connection
probabilities, when appropriately rescaled, have a well-defined and conformally
covariant scaling limit and therefore behave like CFT correlation functions.
While constructing a full-fledged percolation CFT is still an open problem, in
this paper we prove various CFT features of the scaling limit of
two-dimensional critical percolation.
  We identify several connectivity events, including arm-events and the events
that a vertex is pivotal or belongs to the percolation backbone, whose
probabilities have conformally covariant scaling limits and can be interpreted
as CFT correlation functions.
  For some of the probabilities mentioned above, we prove asymptotic expansions
that can be interpreted as CFT operator product expansions (OPEs) and provide
rigorous versions of CFT fusion rules.
  In some of the probabilities mentioned above, we identify logarithmic
singularities, providing the first rigorous confirmation of similar predictions
made in the physics literature and establishing the logarithmic nature of the
putative percolation CFT.
  The latter result is particularly interesting because, while logarithmic CFTs
are more complex and less studied than ordinary CFTs, they have attracted
considerable attention in recent years due to their role in the analysis of
important physical models and phenomena, such as, besides percolation, the
Wess-Zumino-Witten (WZW) model, the quantum Hall effect, disordered critical
systems, self-avoiding polymers, and the Fortuin-Kasteleyn (FK) model.
