‘Publicly’ is surpassed only by ‘subtlety’ as English's most abominable word. But whilst we're lumbered with the latter, for the former we have a fine and upstanding alternative in ‘publically’. Or we would do if the word wasn't so unfittingly maligned—a fate we can, thankfully, spare it with just a brief look at the history of the word and the logic behind its use.
In 1567 the adverb ‘publikely’ first made its appearance. It stayed that way for about a century until ‘publiquely’ and ‘publickly’ arose in the late 17th and early 18th centuries. The form ‘publicly’ didn't make its appearance until 1855, nearly three hundred years after the first form. Sixty-five years later, we shifted once again to ‘publically’. It was Edith Sitwell who gifted it to us.
But in so-called edited English, ‘publicly’ remains standard. Why? Because of the usual pernicious mixture of prescriptivism and ignorance. Spelling became fixed in English when ‘publicly’ was at the zenith in its vogue, and the simple combination of adjective + "-ly" suffix seems eminently logical. But neither spellings nor meanings are ever entirely fixed in language, and moreover ‘publicly’ isn't as logically consistent as it may first appear.
The OED, after stating that "-al" is often used to form secondary adjectives, notes a cornucopia of adverbs that no longer have their "-al" counterparts. The adverb, it says, "is almost always in -ically even when only the adj. in -ic is in current use, as in athletically, hypnotically, phlegmatically, rustically, scenically." And you can add to that asyndeton my own canonical retort of ‘basically’, as in "publicly is basicly an abomination". Please, let publically be.
Everybody interested in lingustics has their own pet hates, and whilst ‘publicly’ is the most prominent of mine, I do have some others that are perhaps less well founded. One is of pronunciation: when ‘secreted’ and ‘appendix’ are spoken in their senses of hide away and section at the end of a book, I prefer SEE-kruht-id and APP-uhn-dix over the usual seh-KREE-tid and app-EHN-dix. Sometimes my tastes even change. I used to prefer ‘dipthong’ as a descriptivist, but now I prefer ‘diphthong’ as a euphonist.
So I think the linguists of the world should unite and form a counterpart to the Plain English Campaign: the Euphonic English Campaign. And its slogan, I suggest with apologies, should be "sounds good to us".