According to the Citations page of the Wictionary entry for uzhe that Dan Bron refers to, the earliest reported spelling to appear in print was ush:
ush 1914, G. B. Stern, See-Saw, London: Hutchinson & Co., published 1931, page 208:
“So long, Nap! Come over the road after your turn to-morrow, as per ush.”
But despite that spelling's lengthy head start, the fall of the house of ush was probably inevitable, given its awkward approximation of most people's pronunciation of the first syllable of usual.
The next four spellings (chronologically) that Wiktionary reports are from 1997 and 1998, are ushe, yooj (three occurrences in 1997, although I reproduce only the first below), uzh, and yoozh, as follows:
1997, Bruce LaBruce, The Reluctant Pornographer, Gutter Press, page 33:
Kitty and Anouk start making out on the pool table as per ushe until a girl comes over and punches Kitty in the nose for spilling her drink on the green velvet.
yooj 1997 September 7, BTH, “ASSC: Chez Paree SF Highlights, 09/05/97”, in alt.sex.strip-clubs, Usenet:
1015pm, dinnertime. P&Ps. Vissi already ate. IRL comes to keep me company. Fave waitress Bridget gets me "the yooj" (my usual). IRL does the "Jailhouse Special" (bread and water only).
uzh 1998 August 29, Evil Prince, “wRASP...makes you go "Hmmmmm......"”, in rec.models.rockets, Usenet:
I have a LOC Graduator with a 38mm motor mount that weighs 16.25 oz. unladen (chute, Nomex, shock cord, the uzh...) According to wRASP, it would fly 2576 ft with an H220 at 400' base alt. at 72F with a BP of 29.92.
yoozh 1998 September 3, infamous, “ASSC: CP 8/29/98”, in alt.sex.strip-clubs, Usenet:
Good thing the kitchen didn't close until 10:30, since we got there about five after. Bubba had his 'yoozh', and I had spaghetti. We spent half the meal talking about the History of the ASS-C World, and the other half talking good ol' fashioned geek talk.
The fact that these early spellings didn't become standard, despite their relatively early start dates is actually a strike against them, in my opinion: orthography isn't like Linnean genus/species designations in that regard. Notably, yooj, which had a flurry of appearances in 1997, is a fairly close approximation to the way many people in southeast Texas pronounce huge, making it more ambiguous than some of the alternative spellings.
Judging from the header spelling that Wiktionary uses, that source seems to prefer uzhe, which it supports with citations from as early as 2007. But that preference is by no means universal. Marielle Wakim, "Here Are the Correct Spellings of the Most Important Abbreviated Words We All Text," in LA Magazine (June 8, 2016) plumps for yoosj:
NO: Yoozh, ush, usj, yousche, ushe, ouge, youghe (GTFO with all of these).
Possibly the most divisive word in the history of abbreves, let us all agree that the answer is yoosj. I acknowledge that picking a winner here is like picking the lesser of eight evils (maybe more, because if one twisted sicko thinks “ouge” is a great stand-in, who knows how many other bastardizations there are). I know, I know—none of the options feel great. Neither does getting my eyebrows waxed, but I do it because I’m half-Greek and half-Lebanese, and so there is no other option. Such is the case with yoosj. Am I going to win the fictional International Phonetic Alphabet spelling bee with that horrid assemblage of letters? Absolutely not—the noise we crave, that “dsj” sound, is the Prince of fricative vocalizations: it is represented in the IPA by a symbol (ʒ), because in the English alphabet it requires an army of helpful consonants (as in “measure” or “beige”). Which means any dream you may have of abbreviating “usual” with elegance and grace is dead. Which means yoosj.
To which I can't help responding, "Bologna."
A Reddit linguistics forum discussion posted ten years ago has drawn 22 comments, of which the highest-voted responses in descending order are [juʒ] (five votes), /juʒ/ [ju:ʒ] (five votes), yewj (four votes), uzh (four votes), yoozh (three votes), usu (two votes), and us (one vote)
So, basically, no realistically usable winner there.
Merriam-Webster Online devotes an annoyingly undated Usage Notes page to the question at "I'll Have the Us(ual): Shortening 'usual': easy to say, hard to spell," but its conclusion—after giving a fair amount of space to yoozh (used by Joe Marusalk in 2008 and by William Safire in 2009), uzhe (used by Mrc Mohan in 2009), and yuzh (used by Paul Turner in 2017)—is essentially "none [or any] of the above":
Welp, we're afraid that the answer is that there is no established shortened form of usual, which means English speakers are really on their own. Our best advice is to use the form that you think best communicates your intended meaning.
Thank you very much indeed for that, MW.
Not to be outdone by Merriam-Webster, I recommend starting by weeding out short forms that aren't typable on a standard keyboard or touchpad, or that overlap ambiguously with common pronunciations of other words. That disqualifies [juʒ], /juʒ/ [ju:ʒ], us, yooj, yewj, and several others. Then I would reject spellings that don't seem especially close to the sound to be approximated (ush, ouge, yousche, youghe, etc.). Then, and only then, I would review the remaining options and follow MW's sage advice to "use the form that you think best communicates your intended meaning."