Throughout his career he maintained an interest in traditional Scottish music, but he also composed in classical style galant forms.
Like many others whose works feature in the Wighton Collection in Dundee, he was a member of "The Temple of Apollo", a secret musical society of composers in London along with Thomas Erskine, 6th Earl of Kellie, John Reid, Charles Burney and others.
The tunes "The East Neuk of Fife" and "The Flowers of Edinburgh", "two classic reel tunes of the Scots fiddle repertory", have been attributed to him - however "The East Noock of Fife" was already in manuscript in 1709, before Oswald's birth, while Flowers of Edinburgh was included in John Walsh's Caledonian Country Dances in about 1737, long before its appearance in Oswald's collection The Caledonian Pocket Companion.
The earliest version of 'Ae Fond Kiss' was closely modelled on Oswald's tune 'Rory Dall's Port'.
[1] It is unsafe to assume that tunes in the Caledonian Pocket Companion, are his compositions unless this is stated explicitly, although some are unknown elsewhere.