What are the similarities and differences? Are there any general lessons or implications?
Wordreference.com (banned) user 'Ume' claims:
I've found "at whim", not "at a whim", in both the LONGMAN Dictionary of Contemporary English and the Oxford Advanced Learner’s Dictionary. What difference is there between "at whim" and "on a whim". Neither of the dictionaries don't say that.
Wordreference.com Senior Member 'Loob' states without proof:
I'd say the difference between "on a whim" and "at whim" is that "on a whim" relates to a single event, "at whim" to repeated occurrences.
Best Answer
On a whim is used far more often than at whim. One hears "on a whim" ten times a week whereas one reads "at whim" and then only rarely.
The phrase "at whim" means "for no apparent reason, randomly, arbitrarily, whenever the impulse arises, or whenever one likes"
The paramilitaries were walking through the village, torching huts at whim.
I cannot kick back and read a good book at whim these days; there's far too much work to be done to the house.
The phrase "on a whim" refers to a particular instance of unplanned, impromptu behavior:
On a whim, I gave her a call and asked her out.
We went to the beach on a whim.