I've always wanted to see Batman race Sherlock Holmes to solve a mystery. However, Batman's equipment gives him an unfair advantage over Sherlock Holmes, so...
The race will be set some time around 1890. Batman time-travels there somehow. The murder to investigate is the crime from A Study in Scarlet, but I don't think it really matters what mystery they're solving. If you don't want A Study in Scarlet spoiled, you might not want to continue reading.
The investigation is of a man lying in a room, without any exterior injuries or signs of struggle, who is dead on the floor. The word "Rache" is written on the wall in blood, however there are no cuts on the dead man's body. The man has a wedding ring on his person. For more information, search for the book on the internet. It should be somewhere for free, where you can read it as an online text.
Batman doesn't have his utility belt or equipment, only a magnifying glass.
Sherlock Holmes has a magnifying glass.
Who solves the mystery first? Possibly more importantly, who apprehends the killer first?
I'm changing my mind. Originally I said Batman would win. However, this is why I think Holmes would win:
Even though Batman is more knowledgable of everything, Holmes matches his level of knowledge in detection. More importantly, we have proof that Holmes is able to perceive more from a crime scene than most--possibly even more than Batman. In this case, while Batman would likely analyze the body, deduce the cause of death was poison, and then realize that the word "Rache" means revenge, I don't think he would be able to solve the mystery just from that information. He would probably look to question the guard who found the body, and continue from there. On the other hand, Sherlock Holmes (in the book) is able to determine the height of the killer, his appearance, and most importantly, his motive, all from observing the foot prints around the crime scene. Just from this first event in the investigation, Holmes is already farther than Batman.
Batman's downfall is that he isn't able to think "backwards." Holmes explains that that is how he solves mysteries so adeptly. Rather than use a process of elimination (narrowing down possible suspects), he discerns the suspect's traits before ever meeting him.