The dog-cart relates to "The Adventure of the Speckled Band" in a relatively unimportant and indirect way.  However, it occurs in such a classic Sherlock Holmes moment that we must ask you to forgive the indulgence of quoting the entire passage, from the beginning of the story when Helen Stoner has just recently arrived at 221B Baker Street:

Sherlock Holmes ran her over with one of his quick, all-comprehensive glances.
   "You must not fear," said he soothingly, bending forward and patting her forearm. "We shall soon set matters right, I have no doubt. You have come in by train this morning, I see."
   "You know me, then?"
   "No, but I observe the second half of a return ticket in the palm of your left glove. You must have started early, and yet you had a good drive in a dog-cart, along heavy roads, before you reached the station."
   The lady gave a violent start and stared in bewilderment at my companion.
   "There is no mystery, my dear madam," said he, smiling. "The left arm of your jacket is spattered with mud in no less than seven places. The marks are perfectly fresh. There is no vehicle save a dog-cart which throws up mud in that way, and then only when you sit on the left-hand side of the driver."
   "Whatever your reasons may be, you are perfectly correct," said she.
   It seems that Holmes follows this signature style of his for every prospective client he meets.  He puts them at ease with some soothing and encouraging words, and then startles them with a bit of detection that invariably makes the subject think that he knows them or has been spying on them.  The primary surprising detail provided in this case is that Holmes tells Miss Stoner that she rode in a dog-cart along country roads.

   The question in our minds, after having it explained to us that it was the pattern of mud on her jacket's left sleeve that led him to conclude Miss Stoner's method of conveyance, is "What is a dog-cart?"  A dog-cart is a twowheeled horsedrawn light carriage pulled by one horse with seats placed back to back.  The 'dog' part of the name comes from the fact that originally the rear seat folded down to make a box for dogs. (Webster's)  Other sources specified that the dog-cart accommodated 'two persons seated back to back.'  We have to assume from Holmes's reference to Miss Stoner sitting on the left hand side of the driver that the concept of having space for only one person in the front seat was not a hard and fast rule in the definition of a dog-cart. 

So what does a dog-cart look like?  As usual, we have Sydney Paget to thank: in all likelyhood, the horsedrawn device in this picture is a dog-cart.