Microspikes, coils and crampons
Microspikes are short steel spikes on chains, held on by a rubber harness that stretches over any boot. They're what you want for icy pavements, car parks and the paths to waterfalls, and they slip off in seconds indoors.
Coil grips (Yaktrax-style) are lighter and cheaper and fine on packed snow in town, but grip less on Iceland's thick, wet ice. Crampons have long rigid points and strap onto stiff boots for glacier ice — glacier hike and ice cave tours hand them out with a helmet and harness.
Where you'll be glad of them
Reykjavík's pavements after a thaw-and-freeze, the paths at Gullfoss, Geysir, Þingvellir and the south-coast waterfalls, steps down to hot pools, and any car park in the morning. Ice here is often wet and glassy, the most slippery kind.
Buying them in Iceland
Ice cleats (mannbroddar in Icelandic) are sold at petrol stations, outdoor shops and some supermarkets around the country, but they cost more than at home, so pack a pair if you have one.