The argument that the word contains a double negative and and is therefore improper ignores countless other commonly used words which do the very same thing; e.g., "debone", "unravel". A similar argument could also be made for the seemingly nonsensical yet true proposition that "flammable" and "inflammable" mean the same thing.
It also fails to take into consideration that even standardized, "proper" English has very few firm rules, that most rules of proper English have many exceptions, and that formal vocabulary, grammar, spelling, etc. vary from place to place. (Colour/color; 'quotes'/"quotes"; "punctuation inside quotes,"/"punctuation outside quotes", possessive's apostrophe/its lack of apostrophe, while/whilst, and the like.)
Of course, this does not excuse the complete and utter abuse of English by many people today; e.g. "u" for "you", "ur" for "your"/"you're", "i" for "I", "wat" for "what", etc. However, "irregardless" does not fall into this blatantly vulgar category.
Husband: "Kiss my ass bitch! I'm still going to the strip club tonight!"
Irregardless is a non-word that many a tool mistakenly believe to be correct usage in formal style, when in fact it is used chiefly in nonstandard speech or uneducated writing. Coined in the United States in the early 20th century, it has met with a blizzard of condemnation for being an improper yoking of irrespective and regardless and for the logical absurdity of combining the negative ir- prefix and -less suffix in a single term. Although one might reasonably argue that it is no different from words with redundant affixes like debone and unravel, it has been considered a blunder for decades and will probably continue to be so.
To be honest, I dont mind people using words like this in normal speech, but I cant stand to see it in print. People like journalists and authors, I believe, have a responsibility to be at least somewhat gramatically correct.
Other person: "Alot of words are considered ungrammatical before they are sanctioned."
Me: "Yes, and it's a sad testiment to the quality of education in society when these mistakes are sanctioned!"