These can be adapted to your needs whether you write a ”prospecting” (unsolicited) letter to firms that have not announced an opening or a “solicited” letter for a publicized opening.
POSSIBLE COVER LETTER OPENINGS
This is a summary of your two or three outstanding qualifications. Like the first paragraph of a newspaper article, the summary gives, in capsule form, the important points you will expand in the message, such as “My classes/past work experiences/volunteer experience has prepared me for. . . .” Then you would use the next one or two paragraphs to talk more fully about the things just mentioned.
If someone such as "Mr. Jones, Vice President of Marketing" has suggested you apply to particular firm, you can use that person’s name (unless s/he has asked you to keep it confidential). The ultimate value of such beginnings depends almost entirely on the named used; but the fact that a friend, business associate, or customer is mentioned invariably wins consideration.
This type of opening has the desirable effect of forcing the applicant to plunge into his/her most salable qualities without slowing down the message by introduction. An opening of "Does your company need a motivated self-starter with substantial training in Microsoft applications" necessitates that applicants using the question beginning be absolutely certain that their qualifications DO answer the question that they raise!
This form incidentally mentions some point of knowledge about the company. The fact that you can state "Your recent 8% increase in local market share" implies a degree of interest greater than that demonstrated by applicants who canvass the general market.
You should, of course, design your own opening—the most appropriate for you and for the particular company to which you are writing. You can state an interest, previous experience with the same organization, a belief, or your present situation.