Context. The various mechanisms of magneto-convective energy transport determine the structure of sunspots and active regions. Aims: We characterise the appearance of light bridges and other fine-structure details and elaborate on their magneto-convective nature. Methods: We present speckle-reconstructed images taken with the broad-band imager (BBI) at the 1.5 m GREGOR telescope in the 486 nm and 589 nm bands. We estimate the spatial resolution from the noise characteristics of the image bursts and obtain 0.08″ at 589 nm. We describe structure details in individual best images as well as the temporal evolution of selected features. Results: We find branched dark lanes extending along thin (≈1″) light bridges in sunspots at various heliocentric angles. In thick (≳ 2″) light bridges the branches are disconnected from the central lane and have a Y shape with a bright grain toward the umbra. The images reveal that light bridges exist on varying intensity levels and that their small-scale features evolve on timescales of minutes. Faint light bridges show dark lanes outlined by the surrounding bright features. Dark lanes are very common and are also found in the boundary of pores. They have a characteristic width of 0.1″ or smaller. Intergranular dark lanes of that width are seen in active region granulation. Conclusions: We interpret our images in the context of magneto-convective simulations and findings: while central dark lanes in thin light bridges are elevated and associated with a density increase above upflows, the dark lane branches correspond to locations of downflows and are depressed relative to the adjacent bright plasma. Thick light bridges with central dark lanes show no projection effect. They have a flat elevated plateau that falls off steeply at the umbral boundary. There, Y-shaped filaments form as they do in the inner penumbra. This indicates the presence of inclined magnetic fields, meaning that the umbral magnetic field is wrapped around the convective light bridge.