Coscinomancy

Meaning: Divination by the sieve and shears. It's difficult to explain, but Brewer claims that:

The points of the shears were stuck in the rim of a sieve, and two persons supported them with their finger-tips. Then a verse of the Bible was read aloud, and St. Peter and St. Paul were asked if it was A, B, or C (naming the persons suspected). When the right person was named, the sieve would suddenly turn round. 

The first image here will give you an idea of the fragility of the arrangement on which someone's guilt might rest.

Usefulness: 2 (Mainly as a synonym for a pointless, arcane ritual that confirms what everyone suspects. Elections, perhaps?)

Logofascination: 3 (I'm more fascinated by the weird and wonderful ways humans have sought for truth.)

Capricious

Meaning: You probably already know the meaning of this word, but Cotgrave defines caprice as:

A humor, caprichio, giddie thought, fantasticall conceit; a suddaine will, desire, or purpose to doe a thing, for which one hath no (apparent) reason.

I rather like the cynicism of that (apparent). Caprichio was a synonym for caprice, but now survives only in music.

Usefulness: 1 (Although if you wish to substitute giddie thought from now on, I won't blame you.)

Logofascination: 1 (It's a goat word! Probably! In fact, according to etymonline it's either from the Latin for wild goat, or a word about people with curly hair, and how can you not love that? I had a lightbulb moment when someone at work used it this week, and unlike a lot of etymological lightbulbs, this one was correct! Probably! Goats have a suprisingly strong influence on etymology, so I'm erring in their favour.)

Mastigophorous

Meaning: whip-bearing; used humorously* of British schoolmasters.

Usefulness: 1 ("Morning boss; looking particularly mastigophorous today." "You should all know that my approach to this year's budget will be mastigophorous."  Can obviously also be applied to jockeys, rodeo riders and other less savoury occupations.)

Logofascination: 1 (From the Greek for 'whip' and 'carrying', it's the sort of word Sir Thomas would have invented.)

Calefaction

Meaning: warming and heating, or being warmed and heated.

Usefulness: 1

Logofascination: 1 (Rabelais has the French échauffer, but Sir Thomas uses the English calefaction, which is from the same Latin root. Perhaps his coinages occurred where he felt he needed an equivalent to the French he was translating.)

Paroemiographer

Meaning: a writer or collector of proverbs.

Usefulness: 1 (I'm considering adding this to my resume. Could also be applied to Twitter, where there are a number of people attempting to find or coin the most applicable proverb of the day.)

Logofascination: 2 (This word gets points just for existing.)

Astragalomancy

Meaning: divination using the huckle-bones (a small bone around the ankle, I think) of sheep; apparently these days people have gone soft and use dice instead.

Usefulness: 3 (Unless you're on a sheep farm. Or in New Zealand.*)

Logofascination: 1 (The name comes from the Greek astragalus, their name for the bone, which also gives us astragaloid and astragalize, the latter meaning to play at dice.)

Enixibility

Meaning: the possibility of being born - the OED says "?* Possibility of being brought forth".

Usefulness: 2 (Could be quite useful in the film industry: "Hugh Jackman's signed on, which automatically triples the enixibility.")

Logofascination: 1 (From enixus, a word to do with struggling which came to mean giving birth.  The OED takes the rather unusual step of listing enixibility as a "nonce-wd. (bombastic.)". The application of 'bombastic' to nonce-words seems to turn up only in a small patch around D and E.)

Disobstetricate

Meaning: the OED says 'to reverse the office of a midwife... to retard or hinder from childbirth'.  Sir Thomas used it in a figurative sense.

Usefulness: 2 (It provides a rather dramatic way of telling someone they'll wish they'd never been born: "I'll make you wish you could disobstetricate yourself!" Could also be used to describe scenes such as Leonato's in Much Ado About Nothing, in which he wishes to disobstetricate Hero: "Grieved I, I had but one? Chid I for that at frugal nature's frame? O, one too much by thee!")

Logofascination: 1 (Due to being invented by Sir Thomas.)

Glabrate

Meaning: to become or to make bald, hairless, or smooth.

Usefulness: 1 ("Did you glabrate this morning, George? Your chin is a little shady." "And how much would you like me to glabrate today, ma'am?" "Time glabrates us all, my friend.")

Logofascination: 1 (related to glabrousness, which I just like saying aloud; from the Latin glaber, which, somewhere back up the linguistic family tree, is possibly linked to glad and glass.)

Celivagous

Meaning:

Cotgrave: Heaven-faire*; heaven-affecting; wending, or bending, towards heaven. 

The Inky Fool: heavenward-wandering. 

Usefulness: 1 (Either in the spiritual sense, as Rabelais, Cotgrave and Sir Thomas use it, or in the sense used in The Horologicon** of a skyward glance to assess the weather. It's not that useful in Melbourne, where the state of the sky has no correlation to the state of the weather five minutes later.)

Logofascination: 1 (One of the lovely -vagous family of wandering words, as previously mentioned in circumbilivagination.)

Sciomancy

Meaning: Cotgrave:

Divination by conference with the shadows of dead men. 

Usefulness: 2 (There's possibly a case to be made that reading authors long gone is 'conference with the shadows of dead men'.  Otherwise this word is probably only useful if you're writing a horror story, or reviewing The Sixth Sense.)

Logofascination: 2 (A specific form of necromancy, etymologically from the Greek for shade and shadows.  There's some suggestion that it can also mean divination by shadow - what shape yours casts, or the movement of shadows. I haven't found much to corroborate either way - what information I could find regarding sciomancy refers back to Rabelais, and I had no particular desire to go digging in the dark corners of the internet for the details of necromancy.)

Nivel

Meaning: among other things, to wrinkle one's nose at someone or something.

Usefulness: 1 ("Don't nivel at me, young lady!" "In my day, there'd have been none of your nivel; we'd have loved a nice bit of brains, my word.")

Logofascination: 2 (I'm a little surprised that this died out; I can't think of another word for this particular movement. It should be pointed out that nivel can also cover grimaces, and may be related to snivelling, or to an old word for brow or forehead.)

Subnovitripartient

Meaning: in a ratio of one to nine and three parts; in the example used by Sir Thomas, nine and three eighths, or 9.375.

Usefulness: 3 (I suspect this word's main use is to impress people; you could, for example, describe the Hogwarts railway platform as the platform of subnovitripartient fourths.)

Logofascination: 1 (Built from blocks of Latin: sub-, used in this case to reverse the ratio; novi-, nine; tri-, three; partient, dividing. The sub- indicates that it's one to nine and three parts, rather than vice versa.)

Myrmecophagous

Meaning: ant eating

Usefulness: 3 (Unless you work in a zoo, the only practical application I've come up with is to describe the suggestive Antz Pantz ad of the 1980s. Although as an insult "How myrmecophagous of you" could mean just about anything.)

Logofascination: 1 (Sir Thomas would approve of its construction: myrmeco-, meaning ant, and -phagous, eating, are both from Greek via Latin. I liked it so much that I resisted the more topical hippophagous.)

Proquiritations

Meaning: the Dictionary of Scots Language has "to cry out in public, publish, proclaim", but for reasons outlined below, I think Sir Thomas meant "to publicly protest on behalf of another".

Usefulness: 1 (Can be used seriously: "The proquiritations of the Occupy movement seem to have made little difference". Or not: "It's always good to see Australia beat England, particularly when English proquiritations regarding umpiring provide an excuse to call them whingers.")

Logofascination: 1 (Quiritations are cries for help, complaints or protests, and I think Sir Thomas has added the 'pro' to emphasise that these protests are on behalf of someone else. The Proquiritations are supposedly written by 32 other people arguing that Sir Thomas should be excused from paying his debts, of which there were, apparently, many.)

Scatomancy

Meaning: divination by excremental examination. Wikipedia alleges that dung beetles were sometimes involved.

Usefulness: 2 (Examination of stool is sometimes necessary medical practice; the Bristol Stool Chart is merely a modern form of scatomancy. I suppose you could use it while reading tabloids: "Let's see what this week's scatomancy predicts for society.")

Logofascination: 1 (There really is a word for everything.)

Naufragous

Meaning: Causing shipwreck.

Usefulness: 1 (Besides reefs and certain parts of the Victorian coastline, this word is terribly useful in a figurative sense. Since almost anything can be wrecked - prospects, plans, hearts - anything that wrecks can be metaphorically naufragous.)

Logofascination: 1 (One of those words that makes me think there really is a word for everything. Related to navicular, of course, but also to the frangere family - so something naufragous is something that renders ships frangible.)

Majuscule

Meaning: big letters, capitals.

Usefulness: 1 (To describe angry text: "I see you're using majuscule to express your outrage." Or as your random fact for the day: "Did you know that minuscule has an opposite?")

Logofascination: 1 (Majuscule and minuscule are both typographical terms, taken originally from palaeography. Minuscule text is the small text, in printer's terms the very small text, and so came to mean very small things in general. They are related to major and minor respectively, although the common misspelling, miniscule, shows the influence of mini.)

Nenuphar

Meaning: Cotgrave:

the water Lillie, or water Rose

Usefulness: 1 (I might need to change my ratings to allow for the beauty of a word.  I'm not sure how useful it is, but it is lovely, particularly the French and Spanish version nénufar.)

Logofascination: 1 (Came to us from Sanskrit and Persian via Arabic nilūfar and a transcription error.)

Soberty

Meaning: Sobriety

Usefulness: 3 (It's an old alternative for sobriety; I just like the sound of it. "I'm rather looking forward to a month of soberty, and so is my liver." "Perhaps if you indulged in soberty more often, your memory would improve.")

Logofascination: 2 (The OED has citations from 1303 to 1483, and the more boring sobriety isn't attested until 1401, after which it seems to dominate.  Sobriety is closer to the Latin root, sobrius, which is not ebrius, not drunk. Ebrius is of course the root of inebriated and the rather lovely ebriety. I quite like the fact that one is, etymologically, in a state of being drunk or not-drunk, drunk therefore being the default.)