Where have fractals gone since Mandelbrot? What are some examples of cutting-edge research involving fractals or self-similar structures?

Who's actively contributing high-quality research in this field?
As pointed out in the comments, this is a tremendously broad area and there are many people doing a lot of very good research, so I can only say a little bit about the small part of it that I'm familiar with.  When dealing with "chaotic" dynamical systems (a notion which can be made precise in a few different ways), one often finds that the chaotic behaviour of the dynamics is intimately connected to the presence of various geometric structures in the phase space that are best characterised as fractals -- the Julia sets found in complex dynamics are examples of this.

To study these structures, one uses various dimensional quantities, such as Hausdorff dimension.  It turns out that some fundamental dynamical quantities such as topological entropy can also be defined as "dimensions" of a sort, so that there are deep connections between fractal geometry and chaotic dynamics.  One of the standard (advanced) references on this is "Dimension Theory in Dynamical Systems", by Yakov Pesin.  A more introductory exposition (with apologies for self-advertising) is "Lectures on Fractal Geometry and Dynamical Systems", by Pesin and Climenhaga.

As an example of the sort of thing that occurs, suppose you have a dynamical system with phase space $X$ and an observable function $\phi\colon X\to \mathbb{R}$.  You take measurements of $\phi$ as time goes along, and calculate its average value as time goes to infinity.  This average value is a function of your starting position $x\in X$.  Let $K_\alpha$ denote the set of points for which that average value is equal to $\alpha$; then typically speaking, there is a range of values of $\alpha$ for which $K_\alpha$ is a fractal with quite an intricate structure.  This is called a multifractal decomposition.  One can define a multifractal spectrum by $B(\alpha) = \dim_H(K_\alpha)$, and it turns out (somewhat miraculously) that the function $\alpha\mapsto B(\alpha)$ is actually concave and analytic in a great many (important) examples!  This multifractal analysis has deep connections to thermodynamic formalism and statistical properties of chaotic dynamical systems; a good reference is "Thermodynamics of Chaotic Systems", by Beck and Schl&ouml;gl.
render 3D fractals based on the original 2D mandelbrot formula, such as mandelbox and mandelbulb + many more - using free open source software called "mandelbulber"