  We study the fine scaling properties of sets satisfying various weak forms of
invariance. Our focus is on the interrelated concepts of (weak) tangents,
Assouad dimension, and a new localized variant which we call the pointwise
Assouad dimension. For general attractors of possibly overlapping bi-Lipschitz
iterated function systems, we establish that the Assouad dimension is given by
the Hausdorff dimension of a tangent at some point in the attractor. Under the
additional assumption of self-conformality, we moreover prove that this
property holds for a subset of full Hausdorff dimension. We then turn our
attention to an intermediate class of sets: namely planar self-affine carpets.
For Gatzouras--Lalley carpets, we obtain precise information about tangents
which, in particular, shows that points with large tangents are very abundant.
However, already for Bara\'nski carpets, we see that more complex behaviour is
possible.
