Small low frequency magnetic field measurement: soft iron or ferrite coil core?
if I should better use a ferrite core or an soft iron core. The
frequency are low, below 20kHz.

Soft iron will produce eddy currents in the presence of the magnetic field you are trying to sense. Those eddy currents will reduce your signal compared to ferrite so I would prefer to use ferrite.

Will I gain in sensibility with a larger core diameter ?

I assume you mean sensitivity; a larger diameter winding will be more sensitive i.e. it will produce a bigger signal voltage because more magnetic flux will be attracted through the cross sectional area of the bigger winding.

I read that ferrite falls appart when we try to cut it with a
conventional saw.

You could pretty much grind ferrite down to small particles and reshape it in any way and get a decent result with glue. It's easily ground down and sand-papered. You could try using a fine grade hacksaw and I wouldn't be that concerned about little chips breaking-off because, the dominant path for magnetism is still via &quot;the air&quot; and, small to medium irregularities in the ferrite won't cause a problem. The ferrite core is just a &quot;concentrator&quot; of flux and it can be any shape and give pretty much the same benefits.

may I assume that the higher the inductance, the more sensible to
magnetic field my coil will be ?

Inductance is a by-product - the sensitivity of the coil is related to number of turns and loop diameter. More turns means more inductance.