Well, it's dependent on how the different search modes work, but generally I would say that a table with 3 columns would work:
SearchType SearchValue Count
Whenever someone does a search, say they search for "Company Name: Initech", first query to see if there are any rows in the table with SearchType = "Company Name" (or whatever enum/id value you've given this search type) and SearchValue = "Initech". If there is already a row for this, UPDATE the row by incrementing the Count column. If there is not already a row for this search, insert a new one with a Count of 1.
By doing this, you'll have a fair amount of flexibility for querying it later. You can figure out what the most popular searches for each type are:
... ORDER BY Count DESC WHERE SearchType = 'Some Search Type'
You can figure out the most popular search types:
... GROUP BY SearchType ORDER BY SUM(Count) DESC
Etc.