How significant are third party candidates in US Presidential Elections?

When the Republican Party's candidate, Abraham Lincoln, won the presidency in 1860, his was a minor party. Now, of course, it's one of our two main parties. But there are and have been many "third parties". Anti-Masons, Free-Soilers, Know-Nothings, Prohibitionists, Populists, Socialists, Communists, States' Righters and Libertarians-- they've all nominated candidates for president in the past. America currently have five nationally organised third parties; Reform, Libertarian, Green, Constitution (formerly the US Taxpayers), and Natural Law. Each …

…candidates would also face a post-election problem if they won the presidency. This would be how to govern -- staffing an administration and then working with a Congress dominated by Republicans and Democrats who would have only limited incentives to cooperate with a non-major-party president would render an Independent President useless. However, it is likely that if an Independent candidate wins the Presidency that he will have enough popular support to fill Congress with Independents.