How deep are the sands on Arrakis?

There's no canon confirmation of the average depth of the sand. The sole indication seen in the original Dune novels are from the death of Liet-Kynes where he narrates the pre-spice mass forming below his feet:

"only a hundred meters or so beneath him; a worm sure to come, but no way to trap it on the surface and use it.".

He goes on to describe this as

"deep in the sand"

but whether he's indicating that it's near the bedrock (e.g. in the same way that we might describe the "depths" of the ocean) is wholly unclear.

Later on, we see that the Fremen have been building sand-dunes that are truly enormous in scale:

With the downwind face anchored, the windward face grew higher and higher and the grass was moved to keep pace. Giant sifs (long dunes with sinuous crest) of more than 1,500 meters height were produced

If you take a realistic scientific approach then there is a maximum depth that the deserts can be.

When sand deposits accumulate they undergo a process called diagenesis, which involves physical and chemical changes in sediments first caused by water-rock interactions, microbial activity and compaction after their deposition. Increased pressure and temperature only start to play a role as sediments become buried much deeper in the Earth's crust.

During the process of diagenesis porosity of the sediment decreases.

The early stages of diagenesis can take place a depths of tens of meters.

As the sand undergoes additional compaction resulting from further accumulation of sand, the sand comes under increasing pressure from overlying sediments. Sediment grains move into more compact arrangements, ductile grains (such as mica grains) are deformed, and pore space is reduced. In addition to this physical compaction, chemical compaction may take place via pressure solution. Points of contact between grains are under the greatest strain, and the strained mineral is more soluble than the rest of the grain. As a result, the contact points are dissolved away, allowing the grains to come into closer contact.

Mechanical compaction takes place primarily at depths less than 1,000 meters. Chemical compaction continues to depths of 2,000 meters, and most cementation takes place at depths of 2,000–5,000 meters.

So you can assume that even if the bedrock was down below 5000 meters the geological processes that turn sand into rock/stone would make the depth of the desert on average about 3000 meters deep.