Not much.

We know very little about the linguistic construction of the Gnommish language. Eoin Colfer is no Tolkien; his novels are valuable more for entertainment than for deep and complex worldbuilding.

• Gnommish is a spoken language. In-universe, it's not just an enciphered version of English with fancy letters, as the other answers here claim. The fact that Artemis is frequently referred to as being able to speak Gnommish debunks this theory immediately. We also have the following two passages from AF and the Lost Colony, both of which clearly show that English and Gnommish are different from each other when spoken:

  • "No, kind sir," said No1, hurriedly cobbling together a couple of sentences from Lady Heatherington. "Prithee sheathe thine weapon. I bring joyous tidings only."

    The skinny human was confounded. He spoke English as well as the next American, but this little runt was spouting some kind of medieval nonsense.

    Kong straddled No1, holding the knife to his throat.

    "Talk straight, ugly," said the man, deciding to give Taiwanese a go.

    "I wish I could understand," said No1, shaking. Unfortunately, he said this in Gnommish. "What I... eh... meanest to say is..."

    It was no good. Quotes from Lady Heatherington that he could generally shoehorn into any occasion just weren't coming under pressure.

  • "You speak Gnommish?" asked the terrified pixie, in case the big one would eat him for not being polite.

    "Yes," said Artemis. "I do, but Butler is not so fluent. So English, if you wouldn't mind."

    "Sure thing. Not a problem," said Doodah, grateful that he still had the tiny spark of magic left in his brain necessary to fuel his gift of tongues.

• The hieroglyphs used are known. Their correspondence to the letters of the Latin alphabet has been canonically confirmed in The Artemis Fowl Files - see image below. They are also used in most of the novels for the secret messages running along the bottom of all the pages. These messages are written in English with Gnommish characters (using an actual fictional language would have made them much harder to solve, and fewer readers would have bothered), but that doesn't mean Gnommish itself is just English with different characters. It's perfectly possible, for instance, to write English using runes of the Elder Futhark, or characters of the Cyrillic alphabet, but both of those symbol systems were created to be used for non-English languages.

  

• We also know a few words, but not many. You mention the example of "D'Arvit", and the quote below from AF and the Atlantic Complex gives us not only another couple of words but also a very minor piece of grammatical knowledge. But again, Colfer was focusing more on telling an entertaining story than on developing background details such as a fictional language.

  Probably some rubber-based solution, thought Artemis, but the base of the mercenaries' paint was not what bothered him - it was the name itself, which was Plunderer, in Gnommish of course. The verb plunder was pronounced ffurfor and the er suffix that changes the verb to a noun has, in Gnommish, the sound fer, which would imply that one is derived from the other. Grammar lesson aside, the pronunciation of the word plunderer was more or less fourfourfour.

• It's written in lines, formerly spirals. The Book, the ancient text which formed the fairies' version of the Bible, was written in spirals, but the modern-day, more technological fairies write in lines just like most human languages. From AF and the Eternity Code:

  Holly placed the mask over Artemis's eyes and pushed the button. Seconds later, the teenager slumped in his chair. Immediately, lines of Gnommish text began to flash across the screen behind him. In the days of Frond, Gnommish had been written in spirals. But reading in spirals gave most fairies a migraine.

  "Commence deleting," ordered Foaly. "But keep a copy. Some time when I have a few weeks off I'm going to find out what makes this guy tick."

  Holly watched Artemis's life being written in green symbols on the screen.

• Interestingly, we know there are Gnommish symbols for numbers as well as letters, but (as far as I know) what these symbols are has never been revealed. From AF and the Opal Deception:

  Artemis's fingers paused over the keyboard. "I can't read the fairy language or numerals."

  "You can, in fact you are the only human who can," said Holly. "You just don't remember. The pad is standard layout. One to nine, left to right, and zero at the bottom."

  "Nine zero nine," muttered Artemis, pressing the appropriate keys. Holly's cuffs popped on the first try - which was fortunate, because there was no time for a second.