When dissolved in water, salt breaks up into sodium and chlorine ions, which combine with water molecules so they cannot easily sink. However, there is a tendency for streams of fresh water to float on salt water and rise to the top. This caused problems for British submarines in the Dardanelles Straits during WW1. Moving from almost fresh water to the denser salt water, they suddenly became more buoyant and rose involuntarily to the surface, making them visible to Turkish gunners on the shore. There are also parts of the ocean where there are pools of very salty water lying on the bottom in such a way as to clearly show the pool to any diver who happens to see it, as though it were a pool on land, so in some circumstances very salty water can sink.

Turbulence, because seawater is, almost, always on the move saltier water is mixed with fresher by wave action and, to a lesser extent in surface waters, by Brownian motion. In Fjordland the annual rainfall is so high (up to 8000mm) that there is a permanent freshwater layer several metres thick that you can drink from sitting over the salt water from the Tasman in the sheltered inlets. Even there this layer doesn't have a clear cut boundary but rather a mixing layer where the salt and fresh water exchange particles and homogenise over time. In bodies of water that don't experience regular circulation stagnation and anoxia set in over time but chemical solution of a number of dissolved salts still occurs.

Saltier water has higher mass density, so the gravitational energy can be lowered that way. The concentration differences go up until the free-energy of creating that big a concentration difference balances the gravitational energy change.
Department of Physics, University of Illinois at Urbana-Champaign

Making some simplifying assumptions, they find:

the equilibrium concentration goes up exponentially with depth, by a factor of e for each 10 km or so.
The actual oceans are stirred by currents, so this equilibrium concentration difference isn't present in them.

Basically they saying that it takes energy to separate a homogeneous solution into parts which are more or less concentrated (and hence more or less dense). Taking into account the gravitational energy, it follows that the least energy state of a column of water is saltier at the bottom.

Why does the salt in the oceans not sink to the bottom?

Because there isn't any "salt", per se, in the ocean. Salt, as the compound sodium chloride (NaCl) does not exist as a solid in the ocean. It is dissolved into sodium and chloride ions (charged atoms) that exist within the ocean as a homogenous phase (that is, a "thing").

That said, water with sodium chloride dissolved in it is indeed denser than pure water, because after all, sodium and chlorine atoms are denser than atoms of hydrogen and oxygen. This leads to an interesting phenomenon: you can have layers of more-salty water and less-salty water that do indeed rise and sink. There are several YouTube videos that demonstrate this very well. For example this video shows dyed salty and fresh water, separated by a barrier:

and then when the barrier is released, the salty water sinks down:

Source.

Some other videos: one and two.

This phenomenon is extremely important for planet-scale ocean circulation, and has strong influence on our climate.

But it does, but according to each salt's solubility and density. Soluble salts tend to mix into the water and keep suspended. Insoluble salts separate from the solution and creates deposits in the oceanic floor.

One famous example was the "de-ironing" of the seas, when iron salts were deposited in the bottom due to the oxigenation of the oceanic water, by the time of the emergence of aerobic, photosynthetic organisms.

Great Oxidation Event: https://en.wikipedia.org/wiki/Great_Oxidation_Event

"The oxygen then combined with dissolved iron in Earth's oceans to form insoluble iron oxides, which precipitated out, forming a thin layer on the ocean floor". https://en.wikipedia.org/wiki/Banded_iron_formation

I'm a regular from the Physics Stack Exchange reporting for duty.

Why this is a serious question

This is a bigger question than you might be giving it credit for. The question is ultimately similar to asking why all the air molecules in the atmosphere do not fall to the floor. Your question comes from a very solid principle in physics which could be called the minimum energy principle.

The basic derivation is that if you define the power exerted by a force $mathbf F_i$ on a particle with velocity $mathbf v$ to be $$P_i=mathbf F_icdotmathbf v = |mathbf F_i|~|mathbf v|~costheta,$$then Newton’s law that the sum of forces on a particle $sum_i mathbf F_i = m~dot{mathbf v}$ is the mass times the change in velocity per unit time, directly implies that the sum of powers exerted on a particle $sum_i P_i = dot K$ is the change in kinetic energy per unit time. Drag forces exist and they oppose relative motion, so their $cos theta$ is negative and they will decrease kinetic energy, $dot K < 0.$ Since energy is a conserved quantity (a “stuff,” if you’d like: if you find more or less of it in a box, then it must have come from somewhere else where there is less or more of it), drag forces eventually rob energy from a system until it ends up at the minimum energy.

And it is a very useful principle, for example you can use it to very easily derive the principle of buoyancy and the effective force that must be created by the displaced water to produce that effect; you can't do Newton’s laws easily when there are that many tiny little forces of little water molecules but you can absolutely compare total potential energy when an object is at the bottom of the ocean, the middle, and the top. It fails to describe certain things like static friction (why is my laptop on my desk and not on my floor?!) because it does not tell you how long such things take and requires an assumption of noise to eventually perturb you out of “local minimums” and such.

But surely the air has had enough time to fall to the ground if that were what it wanted to do. The air does not want to fall to the ground. And we can’t steal our normal solutions for other things like “why don’t clouds fall,” “well what you think of as a cloud is actually more like a waterfall, there is constant movement of water droplets, the water gets a boost upward from heating the air around it as it condenses but it does tend to eventually fall but when it falls beneath a certain flat surface it evaporates again and becomes invisible and so the visible puff is constantly being fed by new water droplet formation and constantly sapped by falling water that becomes invisible…”—no. These are concrete particles that somehow avoid falling to the ground and we have to actually solve the problem.

Fluctuation-Dissipation theorem to the rescue

The minimum-energy principle describes something that we would call dissipation, energy leaving one system to end up in another system. These sorts of gates are always bidirectional: energy goes through in both ways. But mostly you don't notice it, and that’s key to how the principle helps us describe things: energy always flows out, it never flows back in.

Until, well, it does. Energy of a bouncing ball spreads out among all of the different degrees of freedom of the floor, the air, but if it really goes all the way to 0 and sits perfectly and completely still, very soon the air will bump it and start it jostling and vibrating and moving again—just not moving very much. The same things that allow energy to dissipate must also be contributing constant energy fluctuations that prevent energy from going all the way to 0.

These fluctuations are collectively understood as temperature. Temperature is technically only defined for a system where all of its degrees of freedom in the ways it can move have come to the same average energy, and it is measured as that average energy. Temperature defines this average energy and the size of these fluctuations. So at room temperature for example we would say that every degree of freedom has 26 meV, 26 "milli-electron-Volts" of energy, or 0.026 of the energy that an electron would have if accelerated by a one volt battery.

So why does the air stay up? It is, basically, because the molecules of the floor are kicking the air molecules with enough energy to hit the upper reaches of the atmosphere. They do not actually go straight there; one air molecule bumps into other air molecules over a very short distance scale: but it transfers that energy and momentum to other particles which transfer that energy and momentum to other particles and in the end the air "prefers" to "hang out" near the ground but the fluctuations cause it to get bumped to an average height given by our temperature. So if you take the mass of nitrogen N₂ of 28 amu, and the acceleration due to gravity of 9.8 N/kg, you can find out that this 26 meV temperature means that the atmosphere is ~9 km high on average, which does get you a good chunk into the troposphere where the air starts to thin out dramatically. Actually the theory says that if nothing else were to happen and the random kicks were to just launch a particle up into the atmosphere, it would have a random height sampled according to an exponential probability distribution, $P(h) sim e^{-h/(9text{ km})}$.

Similarly why don't the salt molecules fall to the ocean floor? Well, they do, and then they get kicked back up. The water at the ocean floor is saltier. The key difference is whether the salt in question dissolves in water (if it sticks to water better than it sticks to itself) or precipitates in water (it sticks to itself better): larger chunks of a piece of stuff that get bound together will tend to act as big massive chunks and then that thermal energy cannot kick it as high.

This is the general idea of the fluctuation-dissipation theorem, which states that fluctuation and dissipation (under some extremely broad assumptions called “detailed balance”) always go hand-in-hand. Anything which can absorb light (dissipation) must radiate light into space (blackbody radiation, a sort of fluctuation). Every electrical resistor is also a noise source (Johnson noise). If energy can flow out of a system into some environment, then it will only flow out until they have the same average energy levels, and if you try to go lower, energy fluctuations from the environment flow back into the system.

And then there is the saturation issue. Salt can be dissolved in water to a certain degree only. Once that degree is exceeded the salt begins to fall out and sink to the ground. If I remember well the limit for water is something like 35g per litre (depending on the temperature)

Salt does sink to the bottom in the oceans.

Why? Your question referring to salt. Salt is a solid chemical compound. Take a lump of rock salt of sodium chloride, throw it into the water: it will sink to the bottom. The reason is that the density of sodium chloride with more than 2 g/cm³ is higher than the density of seawater less than 1.1 g/cm³.

Of course, the salt lump will be dissolved sometime and no longer exist. But then it's no salt anymore. Then there are only fast and somehow moving loose cations and anions in the water.

Salt does not sink to the bottom in the seas and oceans, because it dissolves in water! If you want to get salt from the seas and oceans, try to vaporize them :-)...

Actually it does. But the thermo-haline cycle of cold water and upwelling mixes that water around. In stagnant stand, the salt will accumulate on the bottom. But ocean currents and the rise and fall of varying temperature water flushes that salt and moves it around.