The Asp was a sloop. According to a chart as of January 1805 in "The World of Jack Aubrey: Twelve pounders, Frigates, Cutlasses, and Insiginia of His Majesty's Royal Navy" by David Miller, the sloops in commission varied from 14, 16 or 18 guns.

"The term "unrated" covered a wide variety of smaller vessels, but while the stately ships-of-the-line fought the great fleet actions and the dashing frigates conducted many glorious actions, it was the hundreds of smaller vessels that kept the whole Royal Navy machine going. These vessels carried out the dozens of mundane tasks, many of them dreary and repetitive, such as inshore patrols and convoy escorts, but without which the naval war would have come to a halt." p.26

Sloops were commanded by a Commander and not a Post Captain.

The Laconia was a frigate. Frigates were fifth or sixth rate (lowest on the scale of the six classes of rated ships), carrying 20 to 50 guns on the main deck. These are the general guidelines, since there are some exceptions. Greater detail is not provided in Persuasion.