I have also found this an interesting question. It's hard to know where to start.
Googling, the best I've done so far is this brief, rather unenlightening entry under the heading "United States Sharpshooters":
Before the outbreak of the Civil War, a few hand picked sharpshooters were used as a presidential guard (On the day of President Lincoln's inauguration Sharpshooters were used because of the threat on his life from the rebellious confederacy).
-from
http://www.angelfire.com/co3/sharpshooters/usss.html
From a couple of books I have about Washington D. C. at that time, I found that Charles P. Stone had been put in charge, and am more inclined to believe that the "sharpshooters" may well have come from a militia unit.
From Wikipedia I found this:
"At the outbreak of secession, Stone found himself in Washington writing his report on Sonora. After a dinner with his former commander General Scott, Stone was requested to be Inspector General of the District of Columbia Militia at the rank of Colonel as of January 1 and was thus the first volunteer officer mustered into the Union Army before the Civil War. In this role, he secured the capital for the arrival of President-elect Abraham Lincoln and was personally responsible for security at the new president's inaugural."
From
Freedom Rising by Ernest B. Furgurson, "To defend Washington, at year's end [1860] there were fewer than 500 regular Federal troops - between 300 and 400 Marines ..., plus 50 or more ordnancemen ... . In theory they were backed by four small volunteer militia companies .... The four together totaled perhaps 500 more men, whose loyalty to the Union was by no means certain."
[One unit that can be discarded would be "National Rifles", one of the four militias referred to above.]
"The day after Charles Stone was appointed inspector general of the capitol, he encountered Captain F. B. Schaeffer, of the National Rifles, ... and complimented him on his company's reputation for precision drill.
"Yes, it is a good company," said Schaeffer, who had not heard of Stone's new commission. "I suppose I shall soon have to lead it to the banks of the Susquehanna!"
"Why so?" asked Stone.
"Why! To guard the frontier of Maryland and help keep the Yankees from coming down to coerce the South!"
Stone ... did not disclose his own assignment. Later, he admitted that "this was not a very cheerful beginning.""
From
Reveille In Washington by Margaret Leech, comes this info about the four militias:
"The four existing organizations of citizen soldiers, three in Washington and one in Georgetown, were already supplied with arms." [The Washington units:] "the Washington Light Infantry and the National Guard Battalion were old and dependable establishments. The third and most fashionable ..., the Washington Rifles, was a hotbed of disloyalty."
[Stone] "ordered Schaeffer to deposit in the armory the howitzers and other unsuitable armament, and the order was obeyed. Eventually, the captain resigned, taking the elite of his men with him to serve in the South." [they formed a company of the First Virginia Infantry, under Schaeffer's command].
That's all for now. I'll do a little more research on some of these other militia units, and continue this post.