We describe an instability that appears at the front of laminar gravity currents as they intrude into a viscous, miscible ambient fluid. The instability causes a current to segment into fingers aligned with its direction of flow. In the case of currents flowing along a rigid floor into a less dense fluid, the case of primary interest here, two mechanisms can produce this instability. The first is gravitational and arises because the nose of the gravity current is elevated above the floor and overrides a buoyantly unstable layer of ambient liquid. The second is a form of viscous fingering analogous to a Saffman–Taylor instability in a Hele-Shaw cell. Whereas the ambient fluid must be more viscous than the current in order for the latter instability to occur, the gravitational instability can occur even if the ambient fluid is less viscous, as long as it is viscous enough to elevate the nose of the current and trap a layer of ambient fluid. For the gravitational mechanism, which is most important when the current and ambient fluids have comparable viscosities, the wavelength when the instability first appears is proportional to a length scale constructed with the viscosity, the flux and the buoyancy. The Saffman–Taylor-type mechanism is most important when the ambient liquid is much more viscous than the current. We have carried out experiments with miscible fluids in a Hele-Shaw cell that show that, at the onset of instability, the ratio of the finger wavelength to the cell width is a constant approximately equal to 2. This result is explained by using the principle that the flow tends to minimize the dissipation associated with the finger perturbation. For the gravity currents with high viscosity ratios, the ratio of the wavelength to the current thickness is also a constant of about 2, apparently consistent with the same mechanism. But, further analysis of this instability mechanism is required in order to assess its role in wavelength selection for gravity currents.