  We investigate six types of two-point boundary correlation functions in the
dense loop model. These are defined as ratios $Z/Z^0$ of partition functions on
the $m\times n$ square lattice, with the boundary condition for $Z$ depending
on two points $x$ and $y$. We consider: the insertion of an isolated defect (a)
and a pair of defects (b) in a Dirichlet boundary condition, the transition (c)
between Dirichlet and Neumann boundary conditions, and the connectivity of
clusters (d), loops (e) and boundary segments (f) in a Neumann boundary
condition.
  For the model of critical dense polymers, corresponding to a vanishing loop
weight ($\beta = 0$), we find determinant and pfaffian expressions for these
correlators. We extract the conformal weights of the underlying conformal
fields and find $\Delta = -\frac18$, $0$, $-\frac3{32}$, $\frac38$, $1$,
$\tfrac \theta \pi (1+ \tfrac{2\theta}\pi)$, where $\theta$ encodes the weight
of one class of loops for the correlator of type f. These results are obtained
by analysing the asymptotics of the exact expressions, and by using the
Cardy-Peschel formula in the case where $x$ and $y$ are set to the corners. For
type b, we find a $\ln|x-y|$ dependence from the asymptotics, and a $\ln (\ln
n)$ term in the corner free energy. This is consistent with the interpretation
of the boundary condition of type b as the insertion of a logarithmic field
belonging to a rank two Jordan cell.
  For the other values of $\beta = 2 \cos \lambda$, we use the hypothesis of
conformal invariance to predict the conformal weights and find $\Delta =
\Delta_{1,2}$, $\Delta_{1,3}$, $\Delta_{0,\frac12}$, $\Delta_{1,0}$,
$\Delta_{1,-1}$ and $\Delta_{\frac{2\theta}\lambda+1,\frac{2\theta}\lambda+1}$,
extending the results of critical dense polymers. With the results for type f,
we reproduce a Coulomb gas prediction for the valence bond entanglement entropy
of Jacobsen and Saleur.
