Is silver magnetic? No. Pure silver and sterling silver are not attracted to a magnet. Hold a strong neodymium (rare-earth) magnet against the item: if it sticks, the piece is not solid silver. Note the reverse is not proof — many non-silver metals are also non-magnetic, so use this test to rule fakes out, not to confirm silver.
A subtler magnet check is the slide test. Silver is diamagnetic, so a neodymium magnet slid down a tilted silver bar slows noticeably as it descends. A fast slide suggests another metal.
The ice test uses silver's exceptional thermal conductivity — the highest of any metal. Place an ice cube on the item: on real silver it starts melting almost immediately, much faster than on a steel or plated piece, because the metal pulls heat into the ice.
Does sterling silver tarnish? Yes. Sterling (stamped 925, meaning 92.5% silver) reacts with sulphur in the air and develops a dark patina over time. Tarnish that polishes away to a bright shine is actually a good sign; a piece that never tarnishes or that flakes to reveal a different colour underneath is likely plated.
Look for stamps: 925, STERLING, 999 or 9999 on bars and coins. No single home test is conclusive — for a valuable item, a dealer's XRF scanner or an acid test gives a definitive answer.