The Red Lantern Corps were started by Atrocitus of Ysmault after his homeworld was destroyed by the rampaging Manhunters, a robotic police force created by the Guardians of the Universe. He swore vengeance upon the Guardians and created the Red Lantern Corps to recruit those with rage equal to his. This led him to conflict with the Green Lantern Corps.
The Sinestro Corps was created by former Green Lantern Thaal Sinestro to enforce the law as he sees fit. He bore a grudge against the Guardians and the Green Lantern Corps, and he inevitably went to war with the Green Lantern Corps. But who has done worse damage to the DC Universe?
10 RED LANTERNS: The Five Inversions
Atrocitus tapped into several unholy magics on his path to creating the Red Lantern Corps. To do so, he became a member of the Five Inversions: five demonic beings that survived the Manhunter massacre in Sector 666. They performed profane rituals and experiments on Ysmault, but Atrocitus felt no loyalty to the Inversions--he only served his rage. Atrocitus killed the other four Inversions and used their blood to create the Red Lantern Central Power Battery and several red rings.
9 SINESTRO CORPS: Killing Kyle Rayner's Mother
On the eve of war between the Green Lantern Corps and the Sinestro Corps, Kyle Rayner's mother died of a mysterious illness. Kyle didn't understand how this happened, but Sinestro had the answer. A member of Sinestro's new army, a sentient virus called Despotellis, infected Kyle's mother with a deadly disease that took her life. This gave Sinestro the opportunity to take the Ion entity away from Kyle, which was Sinestro's plan all along.
8 RED LANTERNS: Killing Abin Sur
Though the continuity changes from time-to-time, it was retconned in around the time of the New 52 that it was Atrocitus who mortally wounded Abin Sur before he crashed on Earth to give his Green Lantern ring to Hal Jordan. Abin Sur had gone to the Five Inversions for information about the Blackest Night prophecy. The Inversions gave the prophecy to Sur, but Atrocitus held disdain for the Green Lantern. He slashed Abin Sur open and left him to die on Earth.
7 SINESTRO CORPS: Replacing The Green Lantern Corps
At a point when the Green Lantern Corps was (once again) on the brink of collapse, Sinestro and his Corps took the role of top police force in the universe. Unfortunately, Sinestro himself has despotic tendencies, and the Lanterns that make up his Corps often share his views of the universe. He used the opportunity to start experimenting on people with a machine he called "the Fear Engine." It took the return of Hal Jordan with a new self-made power ring to depose Sinestro and disband his Corps.
6 RED LANTERNS: Failed Red Lanterns
Atrocitus had many test subjects while he performed his dark experiments with the Five Inversions. Unfortunately for both they and Atrocitus, many of them survived those experiments.
They became malformed things in perpetual agony and called themselves Abysmorphs. They eventually rose up to challenge Atrocitus on Ysmault. Atrocitus' rage was weakened at this point, but he was still able to slay these creatures.
5 SINESTRO CORPS: Enslaving The Weaponers Of Qward
Sinestro himself could not make the countless yellow rings needed to arm his Sinestro Corps, so he went back to the Weaponers of Quard, who created Sinestro's own power ring years ago. He enslaved the Weaponers and forced them to create the yellow rings. Eventually, they would gain their freedom, and a Qwardian calling himself the Weaponer tried to take vengeance on the Sinestro Corps and Green Lanterns.
4 RED LANTERNS: Crimson Dawn
At the beginning of Rebirth, the Red Lanterns came to Earth to harness the rage of humanity. They created rage towers to spread anger like an infection across the surface of the planet.
Fledgling Green Lanterns Jessica Cruz and Simon Baz stood up to Atrocitus and even briefly cured Bleez of her own Red Lantern ring. In the end, Bleez returned to the Red Lanterns, but Simon and Jess were able to repel the Red Lantern attack and cure the rage infection.
3 SINESTRO CORPS: Assault On Coast City
When the Sinestro Corps finally went to war with the Green Lantern Corps, Sinestro knew the place to attack in order to hurt Hal Jordan the most. He and his Corps attacked Coast City, which was once destroyed by Cyborg Superman and Mongul--the act which led Hal down the path to becoming Parallax. Fortunately, this attack on Coast City wasn't as destructive, as the Green Lanterns were able to beat back the Sinestro Corps and save Coast City.
2 RED LANTERNS: First Encounter With The Green Lanterns
At the end of the war, the Green Lanterns were able to capture Sinestro. However, Sinestro made an enemy of Atrocitus during this war. The first Red Lantern rings were sent out to the victims of the Sinestro Corps' rampage, and Atrocitus and a squad of Red Lanterns killed the Green Lanterns sent to escort Sinestro back to his home planet of Korugar. Atrocitus himself took Sinestro back to Ysmault and crucified him on the Red Lantern Central Power Battery.
1 SINESTRO CORPS: Takeover Of Korugar
The Sinestro Corps' most brutal act actually took place while Sinestro was reinducted into the Green Lantern Corps. In Sinestro's absence, Arkillo and the Sinestro Corps took over Sinestro's homeworld of Korugar. They brutally subjugated Sinestro's people. When Sinestro discovered this, he unleashed a terrible vengeance upon his Corps and, eventually, took back control over them.