[For this I am assuming that the ice cube is made of pure water. Results may differ if the ice contains solutes.]

The rate at which the ice cube melts depends on how quickly heat can be conducted from the surroundings to the ice cube. When an ice cube melts, the resulting water (only that from the ice) is cold relative to most of the surrounding liquid, that is, this cold water has less thermal energy that can be transferred to the ice to cause further melting. If that water could be moved out of the way, then warm water would replace it. Thus, the cube will melt fastest in whichever solution (pure/salty/sugary water) has the fastest movement of the cold water away from the ice cube.

The density of water (and liquids in general) increases as temperature decreases (essentially because the molecules have less energy and therefore tend to be a little bit closer to each other).

This means that for the case of an ice cube melting in pure water the cold water formed from the melting ice is more dense than the surrounding water. It will therefore sink toward the bottom of the container and warm water will replace it next to the ice cube. [This is related to buoyancy. An object submerged in a fluid experiences an upward force related to the mass of the fluid displaced, which is related to the densities of the object and the fluid. If the buoyant force is greater than the downward pull of gravity, then the object floats; if it is less, then the object sinks.] However, solutions containing more than a minuscule amount of salt or sugar will have a density higher than that of pure water at any temperature (salt in table here, sugar in plot here).

So, for both salt and sugar solutions the (pure) water from the ice cube is less dense than the surrounding solution; rather than sinking to the bottom of the container, it will remain "floating" on top of the solution and in contact with the ice cube.

At this point, the transfer of thermal energy to the ice cube depends on how well the surroundings can conduct heat (the thermal conductivity). A bit anticlimactically, it turns out that both salt water and sugar water is around 0.6 W/m-K (in both (sugar) cases (salt) there is some change due to concentration and temperature ). Thus, both solutions should conduct heat toward the ice cube at nearly the same rate. (For what it is worth I think the sugar would be a little faster, but again it depends on concentration and temperature of the solutions.)

To sum up a long answer, an ice cube will melt slowest in purified water, and at roughly the same rate for the salty and sugar water. This is because warm water replaces the cold water for the case of melting in purified water but not when melting in either of the solutions. ( This is a good read for related concepts.) [I would also like to point out that there are some answers to the question with only pure and salt water, here. BUT, only the first one is decent; the other two relate to melting point depression due to solutes. This is a concept that sounds similar and relevant, but is not actually related to the question posed.]

That depends on the concentration of salt/sugar in the impure water, as well as the temperature of the water itself.

The more sugar/salt there is in the water, the lower its freezing temperature and the faster the ice will melt. Of course, warmer water will also melt the ice faster.