These phrases have been the subject of considerable dispute for over a century. In respect of'was attacked as an error or affectation by Marsh 1859. Marsh condemned the phrase in spite of its use by Sir Francis Bacon; he also noted it to be frequent in Coleridge, but he thought Coleridge's influence could not account for its currency. Alford 1866 defended the phrase (he used it himself), citing Bacon, among others. Fitzedward Hall 1873 cudgeled both the earlier commentators; he tended to dislike in respect of (he did credit Coleridge for its revival), but he was contemptuous of Marsh's reasons. Ayres 1881 was content to repeat Marsh; Bardeen 1883 quoted Hall. All of this argumentation boiled down to no more than whether of or to should be the preposition following respect.
As inconsequential as this issue may seem, it did not die off with the dawn of the 20th century. Fowler 1926 helped keep it alive by simply suggesting that the phrases are better avoided than used, which advice is somewhat embellished by Gowers in the 1965 edition. Copperud 1970, 1980 contrives to find something ambiguous about in respect to, and joins with Gowers in recommending about instead of any of the phrases—an oversimple recommendation that, if applied consistently, would make nonsense out of many sentences in which the phrases are used. Phythian 1979 goes his own way, prescribing in respect of and not to. With Phythian we have come 120 years and 180 degrees from Marsh.
None of the foregoing opinion has a solid, practical connection to actual usage. Our evidence tells us that all three of these phrases are in current good use. You will notice that our examples tend to come from academic writing rather than the popular press. In respect of has much more British than American use.
Hull remains completely obdurate about not using the word "recognition" in respect of the French Committee —Sir Winston Churchill, Closing the Ring, 1951
... a map of the world in which all principal features should be correctly placed in respect of mathematically determined parallels of latitude —Benjamin Farrington, Greek Science, 1953
This is particularly so in respect of his concept of lateral thinking —Times Literary Supp., 12 Feb. 1970
In respect to is the least common of the three. It is much more frequent in American English than British English:
The most intriguing in the group of paintings is the one which, in respect to its replicas, has figured simply as the Weavers in the modern literature —W. R. Rearick, Johns Hopkins Mag., Spring 1967
... perhaps prejudices my judgment in respect to the perfection of her model —George Santayana, Persons and Places, 1944
With respect to is currently the most common of the three in American English, and it has British use as well:
... four points I made in my first report with respect to vocational education —James B. Conant, Slums and Suburbs, 1961
... creates its share of problems with respect to language —Albert H. Marckwardt, Linguistics and the Teaching of English, 1966
... a more advantageous position with respect to the larger more comprehensive problem —Nehemiah Jordan, Themes in Speculative Psychology, 1968
... our conception of the physical world can be exhibited as a theory with respect to our experiences —A. J. Ayer, quoted in Times Literary Supp., 19 Feb. 1970