Answer: “seldom” versus “seldomly"
enemytosleep wanted to know, "When do you use 'seldom' versus 'seldomly'?" The answer may surprise you.
In fact, the appropriate circumstance in which to use seldomly is actually never.
Webster's does not even include the word in its online dictionary. The Oxford Dictionaries don't have it either, nor does dictionary.com.
We're forced to conclude that seldomly is not actually a word. Why, then, would some people use it? The reason is that English doesn't always follow what might be assumed to be the logical rules of the language.
An adverb is a word that modifies a verb. Typically, in English, an adverb is formed by adding the ending -ly - as we did at the beginning of this sentence by adding this ending to the adjective (noun-modifier) "typical." However, not every adverb includes this ending. A good example of an adverb that doesn't is ... seldom:
Captain Carrot of the City Watch was seldom seen dressed in anything other than a freshly laundered tunic, gleaming helmet and breastplate, and carefully polished boots.
In this sentence, seldom modifies the passive verb "was seen."
It was said that Lord Vetinari seldom listened to actual instrumental or vocal performances. He preferred to read his music.
Here, seldom modifies "listened." Note that seldom is far from the only adverb in English that lacks any -ly ending. Other examples include almost, often, once, never, soon, yet, always, and many others.
Why, then, would anyone use seldomly? A person who was perceptive enough to notice the patterns of the English language and yet had never studied grammar formally might end up using the superfluous suffix for the adverb, particularly if he or she was trying very hard to be correct and proper:
Commander Vimes frowned as the young troll recruit staggered and then struggled to remain upright. "What's wrong with him? Is he drunk?"
"Corporal Slate?" Sergeant Detritus shook his head. "Not dat one, sir. He drinks seldomly."
Detritus, the first troll member of the Ankh-Morpork Watch, sometimes attempts to improve his manner of speech, with decidedly mixed results. A character like Detritus might indeed use seldomly, but unless you're writing dialog for someone with similar issues, you should be using seldom.