For example, let's assume a string 'green apple'. Is there at least one SE (or, maybe 3rd-party site) which provides its statistics on how often do users search this concrete string?
It's highly doubtful any engine will provide actual numbers for this information, but Google Trends does provide enough to get a feel for a term's recent performance. Whether that's useful will obviously depend upon how precise you really want this information to be.
You can also use the Google Adwords Tool
And here's another tool: Google Insights for Search
it helps you to compare search volume patterns across specific regions, categories, time frames and properties.