Plum is an adjective meaning desirable, and it also denotes the sweet, purplish fruit. The adjectival meaning originated as a figurative extension of the fruit.
Plumb is a verb meaning (1) to determine the depth of, to probe, or (2) to work as a plumber; an adjective/adverb meaning (3) exactly vertical, (4) utterly, or (5) squarely; and a noun referring to (6) a weight on the end of a line, used to determine water depth.
Examples
Plum
Urban Meyer is joining ESPN as an analyst less than two months after he surprised the college football world by resigning from his plum job as Florida’s coach. [The Canadian Press]
When James K. Polk won the presidency in 1844, he named Buchanan secretary of state — a plum appointment. [Salon]
Palin notified CPAC organizers Thursday morning that she wouldn’t accept the plum slot as the conference’s keynote speaker next Saturday night. [The Hill]
Plumb
Though plumb in the centre of Europe, Budapest, for me, always casts a hypnotic east-meets-west spell, thanks to its years under Ottoman rule. [Telegraph]
Major American corporations employ a multitude of psychological and scientific analytical tools to plumb the hearts and minds of customers. [Seeking Alpha]
Or maybe these people are just plumb out of their minds. [Southern Chester County Weeklies]