Simply put, none.
Managed images (those created with create/CompatibleImage) are cached in vram, but all the rendering goes to the surface in system memory (the 'master' copy), and, thus, is unaccelerated (rendering is performed by our software loops).
It's different for VolatileImages: they don't have a system memory copy, and the rendering happens to the surface in vram. Currently (as of 1.4.2) only a small number of rendering primitives is accelerated - drawlines, fillRects, etc (some through ddraw, some through GDI and some via d3d) and copying of other accelerated images to a VolatileImage (and, with a flag, alpla blending of translucent managed images, using d3d).
The stuff said above is valid for windows. On unix it's different.
First, on unix opaque managed images don't have a system memory copy, they are stored in Pixmaps, and all rendering is done to them using X11 (except for the stuff X11 can't handle, of course). Whether this rendering is accelerated or not, and whether Pixmap is stored in vram, is driver/Xserver dependent.
1-bit transparent managed images, however, do have system memory copies, similar to the windows implementation, and rendering goes to the system memory copy.
There's tons of other details, you can take a look at this (somewhat outdated) paperhttp://java.sun.com/products/java-media/2D/perf_graphics.html
Basically, the idea is that one should use VolatileImage for images which are modified often (like a back-buffer), and use managed images for sprites and such.