It seems that every time Star Wars introduces a new Sith warrior, they're revealed to wield a new variation on the classic lightsaber. While Jedi tend to stick to the single-bladed, sword-style weapon (with a few exceptions), the Sith wield a far wider variety of weapons. However, this isn't simply a coincidence, but rather a key aspect of how the two opposing groups see the iconic, "elegant" weapon.
While Darth Vader seemingly wields a classic lightsaber, his blade is actually dual-phased, using two Kyber crystals to allow him to manipulate the reach of its blade. Vader's forebears in the Sith Order are generally less traditional. Darth Sidious dual-wields lightsabers, Darth Maul has his iconic double-bladed 'saber, and Darth Tyrannous uses a curved hilt which complements his combat style (indeed, even Rey's Sith form is pictured with a levered lightsaber.) Meanwhile, Star Wars has also depicted dark side adherents using lightwhips, tonfa-stick-inspired lightsabers, and a lightsaber parasol. Famously, Kylo Ren's lightsaber includes a blazing crossguard, though this is because the Kyber crystal which powers it is cracked, necessitating power vents.
The reason Sith so frequently have non-standard lightsabers is spelled out in 'Obi-Wan & Anakin: The Lesson' and 'Darth Vader: The Lesson' from 2020's Star Wars #25, written by Charles Soule and drawn by Ramon Rosanas and Giuseppe Camuncoli respectively. In these two stories, Anakin is shown training with and learning from both his most prominent mentors, gaining both the Jedi and Sith perspective on the lightsaber. To Obi-Wan, Anakin begins to pitch a new idea for using lightsaber technology to create more varied weapons, but his master stops him in his tracks. Obi-Wan confesses he had similar ideas as a padawan, and suggests that many Jedi-in-training have similar notions about creating unique weaponry. However, he stresses the symbolic meaning of the lightsaber, and how the demands it makes of skill and precision enforce a Jedi's intentionality, making sure they always fight in a deliberate, considered way rather than simply cutting loose.
This lesson is reversed later in life as Vader faces Palpatine, who defeats him in a duel by relying on his Force abilities. Palpatine mocks Vader for treating the lightsaber as significant, claiming that the Sith's real weapon is the dark side. Palpatine believed the only real function of Sith using lightsabers was to belittle the Jedi's symbolic use of the weapon, hence the Sith tradition of forming their own lightsabers by claiming those of defeated Jedi. Taken together, these stories imply that many Jedi padawans have early ideas about creating their own lightsaber-types but are prevented from doing so by Jedi philosophy. Those who subsequently fall to the dark side are then encouraged to warp the traditional lightsaber in mockery of the Jedi, and especially to create superior arms that place less focus on discipline and intent.
This difference in philosophy towards lightsabers speaks to the flaws of both the Jedi and the Sith, as the Sith encourage their adherents to opt for power over self-control, and the Jedi enforce a rigid philosophy that can lessen combat effectiveness. Indeed, the Jedi lock away creative variations like the lightsaber rifle in their vaults rather than finding ways to use them. Interestingly, non-standard lightsabers could suggest a Jedi's propensity to fall to the dark side, speaking to an expression of ego and individualism the Order warns against.
Ultimately, the Sith have more unique lightsabers because the Jedi actively discourage this practice, which many padawan experiment with early in their training. At the same time, the Sith desire to mock and subvert Jedi symbolism is perfectly expressed by taking a traditional lightsaber and warping it into something even more deadly, even if these new forms often come with specific disadvantages. Hopefully, as Star Wars continues beyond the sequel trilogy, a less dogmatic Jedi order will learn from the Sith, allowing padawans to experiment with new lightsaber designs without conflicting with the Order's philosophy.