Plain water - can't hurt can it? Providing that it is in the form of a soft, clean, damp cloth, then most types of jewellery are safe to gently wipe. Avoid soaking in water anything that is glued together (in case a water-soluble glue has been used), strung like beads (because the string inside the bead can rot or the string can stretch)...............
Mild soapy water [coming soon]
Mr Muscle (and other kitchen cleaners) [coming soon]
Washing up liquid [coming soon]
Toothpaste - NEVER. It is abrasive - that's how it cleans your teeth - so it will scratch the metal
Jewellery dips - a liquid you dip your jewellery into to clean it. Some are aimed at silver jewellery; some at all types of jewellery (with strong warnings about use on certain types of jewellery). These contain a weak acid which reacts with the tarnish (or oxidation) on the metal to remove it - and releases a disgusting rotten egg smell in the process!
The problem with jewellery dips is two-fold; firstly they contain ACID which is harmful to all organic gems (pearls, coral, amber, jet, ivory, etc) and also damages many porous gemstones, and those containing copper, such as malachite or turquoise. The acid will also remove decorative oxidation (this is particularly used with silver to blacken the crevices in a jewellery piece and give it an antiqued look).
I have also noticed that rinsing with warm water, as recommended, does not totally remove the cleaning product. When you have dipped an item several times, you will notice a "bloom" or surface discoloration appearing on silver. This gives a milky or yellowish appearance to the surface of silver. If this happens to a piece of plain silver jewellery, I have found that the foil & soda method (see below) will often remove this surface coating.