Helsinki has some docked-bicycles which cost €20/year if you take journeys of 30 minutes or less. They seem to appear in spring and disappear in autumn, so no cheap bicycles in the winter.
That said there are a _lot_ of people who continue to cycle even in winter (-25'C is coldest I've experienced over the past three years), and the snow/ice doesn't seem to stop them. Though of course fat-tyres, and ice-studs are a given.