In American English, catalog is the more common spelling of the word meaning (1) an itemized list of offerings, and (2) to make an itemized list. Catalogue is the usual spelling outside the U.S. A similar U.S.-versus-everywhere-else distinction applies to the spellings of analog/analogue (though both these forms exist in American English and have differentiated in meaning), but other words ending in the silent -ue (dialogue, monologue, epilogue, etc.) have not changed.
Catalog is inflected cataloging and cataloged. Catalogue becomes cataloguing and catalogued.
The following ngram graphs the use of catalog and catalogue in a large number of American texts published in the 20th century. It suggests that the move to the simpler spelling has occurred gradually through this period and is still not fully engrained.
Although the shift to catalog has happened only recently in the U.S., the spelling is actually many centuries old. The original English spelling, when the word entered the language around the 15th century, was cataloge, and there are documented instances of catalog from as long ago as the 16th century. The influence of French, where the word had long had the -ue ending, took hold soon thereafter, and by the middle 17th century catalogue was the preferred form in English by a large margin.
Only in the late 19th century, with the movement in the U.S. to develop a uniquely American form of the language, did catalog begin to make a comeback. It was not one of the original American spellings, though; the word is listed as catalogue in all early editions of Noah Webster’s dictionary, which played a large role in conventionalizing many other uniquely American spellings. By the 1890s, however, instances of catalog are easily found in all sorts of American texts.
The company will continue to produce print versions of its other seasonal catalogs. [Wall Street Journal]
The catalog is filled with hand-drawn illustrations (shown above), color photographs and articles from the 1800s. [Los Angeles Times]
Until the mid-20th century, the world’s greatest artworks were cataloged in shoeboxes. [New York Times]