The safe defaults
Soft natural-bristle brush. Distilled water (tap water leaves mineral residue). A jeweler's loupe or USB microscope to inspect. Work over a soft towel. Brush gently in the natural direction of any fragile crystal faces. Air-dry; never towel-dry. This handles 80% of dust and clay without risking damage.
Ultrasonic cleaners
Useful for hard minerals (quartz, beryl, topaz, garnet, corundum) with NO cleavage and NO fluid inclusions. Fluorite, calcite, soft sulfides, soft phosphates, and anything brittle or cleavable should NEVER go in an ultrasonic. The vibration propagates through cleavage planes and shatters them from inside.
Acid cleaning — proceed with extreme caution
Oxalic acid (used dilute, in distilled water, at low temperature) removes iron-oxide staining from quartz and silicate matrices. NEVER use it on calcite, fluorite, dolomite, aragonite, malachite, azurite, scheelite, or any mineral containing carbonate or fluoride — they dissolve. HCl is reserved for professional preparators on quartz specimens. Acid mistakes are irreversible; if you're unsure, send the piece to a paid preparator.