The SEC is proposing a new rule on board nominations for large publicly-traded companies. Shareholders, or groups of shareholders that control over 1% (3% for mid-sized companies and 5% for small) of the company's voting shares can nominate directors to corporate boards in the company's proxy materials.
The rule still prevents "nuisance" shareholders from making board nominations. Shares must be held for one year to qualify. The cutoff for the 1% rule is $700 million of capitalization, and the cutoff for the 3% rule is $75 million of capitalization. Bottom tier publicly held corporations average about $14 million in market capitalization. Typically, 1% control would require an investment in the tens of millions of dollars or more. The devil is in the details, however. The shareholders who are best in a position to offer competing nominees for corporate boards are institutional investors. But, it isn't clear if they will feel that it is appropriate or useful to use their power to do so.
Under the current regime, just about the only time there are contested elections for seats on corporate boards is when a corporation has been taken over or a large minority of shares of the company have been bought in a tender offer.
UPDATE: There were about 22,000 publicly held companies in the U.S. the last time I looked it up. There might be as many as 30,000 now. In 1971, about 575 companies had 10,000 shareholders or more, about 73 companies had 100,000 or more shareholders, and 2 companies had 1,000,000 or more shareholders (General Motors was one of them). For all practical purposes, every corporation with at least 500 shareholders must be publicly held. Of course, the number of shareholders with 1%, 3% or 5% stakes in the company are much smaller than the total number of shareholders.
More data is available here. In practice, the new rule means that small coalitions of institutional investors can force proxy fights, as can very large private investors (even in bottom tier publicly held companies, a stake of about $700,000 in a corporation would be required, in large publicly held companies, a stake of hundreds of millions of dollars or billions of dollars is required). By comparison, under existing law, existing board nominated board of directors candidates always win (citing this article) (see also here and here).