The Space Corps are the military force, including marines and fleets, which deal with conflicts and peacekeeping away from Earth. These are either Judge Troopers, enlisted citizens, or both depending on the story you're reading.
The main force depicted is Mega-City One's, but foreign Space Corps have been mentioned and shown:
- Sino-Cit in "The Corps" (who have an alliance and co-administered forts with the Klegg Empire)
- Latin American and "Asian Union conscripts" in Warzone, the latter involving an international coalition
- Brit-Cit and Hondo referred to space forces in "Lawcon", when they agreed to join a coalition against the Law Lords
The look of the Corps is entirely dependant on the artist.
They have been primarily shown fighting alien menaces off-world, starting with the Klegg Empire in "The Corps". How often they're fighting invading aliens versus wars of human conquest or clashes over territory is variable. "Warzone" showed them being used as an occupying force on a colony planet to prevent it gaining independence, constantly fighting against local insurgent groups.
They are unlicensed to act against citizens in Mega-City One (Chaos Day riots being the rare exception) and the strategic interests of the city come above the interests of individual colonies, who may be abandoned to alien attack.
When Space Corps soldiers are severely wounded, they will sometimes be turned into cyborgs: nicknamed "Mandroids", they are highly powerful and designed solely for combat. This was a practice already by 2074. 
Corps officers and NCOs are allowed to be married.
The Space Corps was operating by at least the 2070s. They would develop Special Squads from Academy of Law cadets: those showing abnormal levels of violence at ages 3–6, who were trained solely to kill.
In 2109, there were heavy losses in a colony war (not on a MC-1 colony) and citizens were ordered to give blood.
A corrupt platoon of "STAR Judges" committed an act of deliberate genocide against the Quayaan species in 2111. Chief Judge Thomas Silver had the platoon killed for their crimes and then hushed it up, deciding to exploit the situation: the team were 'martyrs', the Quayaan killed in a legitimate act of war, and the planet was rightful spoils of war. 
The Corps fought the Klegg Empire across a dozen worlds in the Outer Spiral Arm in the 2110s, under the command of Colony Troop Leader Keitel in the JDSS Solomon, and by early 2116 they'd lost almost 250,000 men with barely any gain. These were predominantly "colony boys" and many Judge Troopers were invlved in the fighting. Shortly after Hadrian Volt became Chief Judge, the Corps were ordered to start a Sino-Cit/Klegg war through a false flag attack but the operation was a failure.
The Corps fought the Antares Wars for years during the 2110s.
The 45th Infantry ("Legion of the Damned") fought to conquer and colonise Melanndos, but the tribal Pat'u'uk proved able to both shapeshift and turn humans into one of them via infection; they waged a savage guerilla war. Major John Gris was captured for thirty days and mutilated by the "Spooks", and was medically discharged back to Mega-City One in 2118. Shortly after, the Corps abandoned Melanndos as too much trouble. In 2119, Gris - under a psychotic delusion that random people were Pat'u'uk in disguise - murdered dozens of people before Dredd shot him.
The Corps ran a genetically modified soldier program in the 22nd century: soldiers were grown in tubes and indoctrinated, similar to Judges (and especially Black Ops Division but with animal DNA included to enhance them. The program was experimental and one trooper, Armon Gill, tried to eat a civilian.  He was dishonourably discharged in 2122, after twelve years service. This was separate to the Genetic Infantry soldiers, who could be transferred to Space Corps.
Dredd's clone Dolman joined the Space Corps in 2126 but left after 2134.
Sergeant Nate Slaughterhouse was almost killed when his armoured-suit platoon was caught out in the open without air support. His wife, Captain Kitty Rosson, was able to bring him back to base and gave permission for him to become a mandroid - a mistake, as he fell into deep depression. The two resigned their comissions and went home to Mega-City One, where they'd run afoul of their block mafia. After Rosson was abducted & lobotomised and their child killed, Slaughterhouse went on a vengeful rampage and decapitated the gang.
In 2128, an international coalition was trying to stop the insurgency on colony world New Sylvania. Many foreign forces sent conscripts and dregs, while the MC-1 Corps were ill-disciplined; the bitter troops nicknamed this "the Forgotten War", as nobody back home wanted to know about Space Corps fighting humans. Sergeant Duke, head of a Mandroid squad, told Dredd it was inevitable the planet would get independence. General Agers was revealed to be taking bribes from the insurgents and arrested by Dredd. 
The retired (and wealthy) General Trig and a group of other disgruntled veterans attempted a coup in 2129. They manipulated Nate Slaughterhouse into acting as their agent by pretending to restore his wife's mind (actually Trig's terminally ill daughter remote controlling Rosson's body). Dredd would expose the plot and Slaughterhouse died taking out his former allies. 
Hardey Wegg was a hero of the 9th Space Corps for his service on Abraxos, but left with severe trauma & night terrors. He ran for Mayor of Mega-City One - unfortunately, it was the same year PJ Maybe did, and Maybe had Wegg framed for murders and then killed.
After Day of Chaos, Space Corps soldiers were recalled to Mega-City One to make up for the amount of dead Judges. However, in their first mission Colonel Lynn Easter tried to airstrike a dissident city block in a fit of pique and were forced by Dolman to call it off. When news of this got out, the citizens no longer trusted or respected the Corps - and under judicial command, they were forced to hold back on lethal force. The Corps openly saw the Judges as weak.
The alien Zhind raided MC-1 colonies and had small scale conflicts for years. When K Alpha 61 was invaded, the Space Corps did not respond due to the political risk of causing a large-scale conflict with the Zhind.
In April 2136, a crack team of marines was sent under Dredd's command to Titan; during planning, Chief Judge Hershey remarked that the city no longer had the ability to send large forces and Aimee Nixon taunted Hershey that the bulk of their space forces were tied up with ex-Colony Marshall Karel Luther's insurgency, which was seeing apes, mutants, and robots rebelling against Mega-City control and establish independent worlds. Ironically, Luther's driving motive was K Alpha 61 being abandoned.
In 2138 the Zhind launched an all-out invasion of human space across the trilateral line. The Space Corps and other MC-1 military forces had to team up with Luther's insurgents to have the edge. The Corps fought a sixteen-week war against the Zhind, with six major engagements and several minor ones like the planet Malfaize; most of these were fleet battles, including a significant battle at Wolfenstar 77154, but the crucial battle was a land engagement at 43 Rega to stop the aliens reaching the Inner Colonies and the solar system. After the insurgents had broken the Zhind at the Battle of 43 Rega, the Special Judicial Squad's space fleet turned and massacred them.
(It's unclear if foreign Space Corps were involved in the Xhind War as well.)
In 2140, a squad of veteran Marines under the psi Rafael "Rafe" Sabitini was dispatched to the colony world Dominion to stop the Dark Judges. Unfortunately, Sabitini had been traumatised at the battle on Malfaize and his decision to execute wounded soldiers - this left him open to contact from the Sisters of Death and he smuggled Judge Fear on board. The Marines found Fear and a transformed Sabitini, Judge Whisper, manifest before they reached Dominion. While most of the Marines were killed, they were able to blow up Dominion, seemingly destroying Whisper and leaving the remaining Dark Judges stranded once more.  Whisper, sadly, had survived: he got onboard their vessel, killed the remaining Marines, and followed Death to his new location.
The idea of offworld soldiers had been mentioned in 1980s strips, as part of the planning for an aborted Judge Dredd Fortnightly. This comic would have had a spinoff called B.A.D. Company - a sci-fi update of Wagner's old Darkie's Mob. (This was later picked up and drastically revamped by Peter Milligan, though the first part in Prog 500 would follow the basic plot of Wagner's draft) In this, civilians had joined the 1st Mega-City Volunteers to fight the Colony Wars. Former Titan inmate Kano was put in charge of a colony "wipe-out unit", Battalion Assault Deployment Company, that operated permanently behind enemy lines.
The concept was first named and fleshed out in a published strip by Garth Ennis in 1994, as "the Corps", "the military arm of Mega-City 1's Justice Dept". This was written at the request of editor Alan McKenzie, who wanted a war story featuring space Judges; Ennis lost interest after the strip was partially rewritten by an uncredited figure (presumably McKenzie As The Corps has the characters trying to undermine the Sino-Cit/Klegg alliance, it may have been part of McKenzie's push for a storyline (later dropped) where Mega-City One and Sino-Cit would go to war.
Coincidentally, The Corps came out at the same time as a long-delayed, similar strip called Maelstrom, which featured another team of similarly-armoured soldiers called "STAR Judges" (Special Target Attack Retaliation) and also had art by Colin MacNeil. These seemed unrelated but the 2000 AD ABC videos refered to them as "Space Corps". While not stated in the strip itself, the 'Background' pages for Maelstrom stated that the STAR Judges were "a united space force composed of Judges from different Mega-Cities" - characters were from MC-1, Brit-Cit, Euro-City, and Texas City, with Hondo technology. They operated out of the satellite Justice Matrix, which controlled "Justice System, the judicial forces that patrol Earth's offworld colonies", as well as having operating as a conquering military force offworld.
In the later, post-McKenzie stories under John Wagner (renaming them "Space Corps"), the organisation are not Judges but were recruited from the civilian population; military veteran characters would be brought into fiction. This has become a default for later writers like Michael Carroll and Matt Smith
The Mongoose RPG merged the takes and says the Space Corps are "an army of Judges and civilian soldiers".  This was the view taken by 2000 AD ABC, which simplified things by refering to all the space war tales as "Space Corps" and "Space Judges".
Insurrection and Lawless by Dan Abnett present the Special Judicial Squad as having its own, unrelated (and quite Warhammer-esque) space force.
- #240, "Warzone" page 2
- Megazine 334
- Part 1 of The Corps
- 2000 AD #1453-1464
- 2000 AD #1555-1566
- Megazine #240-3
- Prog 1779
- Judge Dredd Megazine #334: "Insurrection III Part 1"
- "Mandroid" stories, "Warzone"
- Machineries of Hate chapter 9
- Year One: City Fathers and Year Three: Machineries of Hate, both by Matt Smith
- Prog 919, "The Corps Part 2"
- Prog 519
- "Maelstrom", Megazine 2.73-2.80
- Prog 918-9
- Prog 923
- prog 1058
- Prog 1263 and 1342
- Prog 1247: discharged "last year"
- Megazine 240, prog 1317
- Megazine 240-3
- Mandroid: Instruments of Terror
- The Gingerbread Men
- Prog 1793-6
- Prog 1801-2
- Megazine 240
- Megazine 310
- Prog 1863: "Titan Part One"
- Insurrection and Insurrection II
- Megazine's recap segment for Insurrection III stated it was 2135 but the beginning takes place simultaneously with the end of Insurrection II, which was recapped as 2133; the strip itself never mentioned years and Abnett said in the "Insurrection: Liberty" trade that he saw this strip as being in Dredd's future. Aimee Nixon refers to the "Luther rebellion" in the colonies in Titan, set 2136, and the 2138-set Megazine #373 referred to the Zhind War in the past tense. "Dark Judges: The Torture Garden" gave the year specifically was "2128", a typo for 2138
- Judge Dredd Megazine #340-2: "Insurrection III"
- Megazine 405
- Judge Dredd Megazine #340-2: "Insurrection III"
- Dark Judges: The Torture Garden
- Dark Judges: Deliverence
- Thrill-Power Overload! by David Bishop
- Reprinted in Megazine 4.15
- Prog 919, "The Corps Part 2"
- Thrill-Power Overload! by David Bishop (page 178)
- Garth said "Alan or whoever" in his full interview with Bishop but that was edited for space, see here
- Thrill-Power Overload! by David Bishop (page 164)
- Megazine #239: "15 Years, Creep! Part 4": the strip had waited a year "for the right artist"
- The 2000 AD ABC: The Corps
- Megazine vol.2 #75
- Judge Dredd core book for the RPG