Batman/Holmes
The protagonist is the most obvious. Both are detectives with cognitive skills far outmatching their contemporaries, both have prominent psychological defects (Bruce Wayne derives his strict moral code from the witness of his parents' murders at a young age and is constantly traumatised by these events long into his adult life, while Holmes' remarkable skills come at the expense of a near-complete lack of human emotion, allowing every living being to be viewed as a puzzle to be solved, and every crime a game that he thrives in and even enjoys, regardless of the seriousness.) Each character operates at the behest of the law, despite the law's reluctant pleasure that they do so, and the exploits of each see them rise to fame in their respective cities. Despite being famously lauded as heroes, both Holmes and Batman possess a somewhat dubious morality. Holmes has little interest in personal justice for the victims of the crimes he investigates, each factor is just a puzzle piece. He shows, for example, no remorse for inadvertedly causing the death of the antagonist in 'The Speckled Band', and on occasion withholds evidence from the police and takes the law into his own hands. He allows the thief in 'The Blue Carbuncle' to go free, simply because it is Christmas. Batman, while holding a strict sense of right and wrong and vowing never to kill, is hubristic in regarding himself above the law and exacting justice without the consent or approval of the police. His methods of apprehending criminals, meanwhile, are somewhat hypocritical in their brutality.
Robin/Dr. Watson
Not immediately equivalent; a Holmes story would be almost unthinkable without the presence and perspective of Dr. Watson, (although they do exist), whereas a Batman story without Robin is not only reasonable but increasingly common, especially among writers that view the character as detrimental to the franchise's darker, adult tone (Robin has not appeared in a 'Batman' movie in fourteen years). However, in each case, the character's initial purpose was to align the story with the reader where the main protagonist is to abstract or austere a character to immediately identify with. As a comic with a strong child readership, Robin makes for an invaluable audience-identifier in the earlier Batman stories, while Watson in the Holmes canon allows for the reader to see Sherlock with the same admiration and awe that he does. Furthermore, each character expands on the protagonist's remarkable abilities by bringing their own to the table. Dick Grayson is a phenomenal acrobat, able to spring Batman from traps and tackle enemies with agility and speed, while John Watson is an accomplished army doctor, often identifying medical anomalies that even Holmes occasionally misses.
Alfred Pennyworth/Mrs. Hudson
An obvious comparison to those familiar with both franchises. Alfred is the butler, seldom seen outside of Wayne Manor in the comics, whilst Mrs. Hudson is the landlady, seldom seen outside of 221b Baker Street in the stories. Each character acts as an iconic piece of living furniture in their respective famous addresses, but they are not merely sympathetic drink-dispensers to their protagonists. While Alfred and Mrs. Hudson often remain separate to the principle story, they are always on hand to aid in their own way. Mrs. Hudson tends to Holmes in 'The Dying Detective' and even assists in his scheme in 'The Empty House', moving the wax dummy of Holmes in the window in order to portray it as living and trick the assailant. Alfred, meanwhile, often assists Batman from the confines of Wayne Manor, conducting extra research and mending damaged gadgets when the protagonist can not. Above all else, each character is the spiritual parent of the franchise, a human signifier of comfort and safety. As Benedict Cumberbatch's Sherlock remarked in the most recent episode of the BBC series, echoing the sentiment of Holmes buffs throughout history, 'If Mrs. Hudson left Baker Street, England would fall.'
Commissioner James Gordon/Inspector Lestrade
The reluctant professional admirer of the protagonist, and by and large the sole link between law and vigilante. Gordon and Lestrade are almost identical in that they serve as the only sympathetic member of the police force and recognise the benefit of Batman/Holmes in crime solving. Lestrade is perhaps treated with less sympathy in that Holmes and Watson continually mock his pedestrian abilities when compared to Sherlock's remarkable skills, but like Gordon, he is unafraid to express his occasional frustration with the protagonist's unorthodox methods of detection. They are dwarfed by the titular heroes, yet Gordon and Lestrade are continually alluded to being the best of their kind. Gordon shines through in a force of corrupt and ineffective police officers, while Lestrade is remarked as being the most competent detective amidst Scotland Yard's bumbling elite.
Catwoman/Irene Adler
The most popular female character of each series (even Dr. Watson's wife Mary is largely absent beyond her first appearance), and both antagonists that are nevertheless held in a light of respect by their respective protagonists. Adler possesses an intelligence equal to that of Holmes, and remains the only character in the canon to successfully outwit him, while Selina Kyle's athletic, stealth and combat skills are an easy match for Batman's. Both characters cause conflict in the morality of the central character by possessing a trait that completely conflicts with them (the painfully mysoginistic Holmes admires 'the woman' Adler despite her sex, judicially-minded Bruce Wayne enters into a romantic relationship with Selina Kyle despite her proffession as a criminal). While Holmes has no truck with romance, there is proffessional admiration for Irene in place of Batman's actual romantic feelings for Catwoman.
The Joker/Proffessor Moriarty
The iconic foil. Neither Holmes nor Batman would be complete without their ultimate adversary. Both serve as the opposite side of the same coin, and both utterly match the abilities of the hero. They operate ominously from off stage, testing the limits of the hero, then finally reveal themselves for a physical showdown with devestating consequences. While the Joker is a constant presence in 'Batman', Moriarty only actually appears in one Holmes story, although Doyle later retroactively made him a presence in the shadows, orchestrating crimes from afar in 'The Valley of Fear', a story set before but written after Moriarty's defeat. Clearly Doyle recognised the character's impact as arch enemy enough to increase the prolificacy of his crimes. Furthermore, Moriarty's absence from the later stories is as keenly felt as his presence in the earlier, and Holmes makes numerous references to his departed foe right up until the very last story, 'His Last Bow'. There is a keen reversal of roles in that, while the Joker recognises how much he enjoys his conflicts with Batman, it is Holmes that continually remarks how much he misses the thrill of the chase with Moriarty, remarking that crime in London has become uninteresting since his death.
There are also some comparisons to be made among the lesser recurring characters;
Bane/Colonel Sebastian Moran
Both criminals that arrive on the scene with the express intention of murdering the protagonist. Moran wants to exact revenge upon Holmes for the death of his boss, Moriarty, while Bane wishes to kill Batman to remove Gotham's overwhelming fear of the Dark Knight. Each possess violent skills that cause the hero to fear for their life (Bane is stronger than Batman, Moran is an expert marksman).Tim Drake/Stanley Hopkins
Tim Drake, as successor to Dick Grayson's Robin, shares similarities with minor 'Sherlock Holmes' character Stanley Hopkins, who is mentioned as being a young police officer with keen detective skills. Hopkins appears in later stories, where Watson spends increasing amounts of time away from Holmes, and is vaguely alluded to being a possible successor to Sherlock Holmes.
Gotham/London
It would be wrong not to mention the respective cities in which the great detectives solve their crimes. Each city is vivdly realised in the fiction, thriving and complex to the point of being something of a character itself. The reader is immersed in this artificial world (while London has the obvious advantage of being a real place, the London of 'Sherlock Holmes' is unique in its stark descriptions of gas-lit, foggy alleyways, rolling hansom cabs and ominous shadows. It is as exclusive to Sherlock Holmes in that respect as the fictional Gotham City is to Batman). Each character is completed, given a degree of dimension, by the city they live in.
