In the BBC series Sherlock (and indeed in most other adaptations of the character I've seen) Sherlock is not paid for this investigations - he does it because he's interested. He rents a flat on Baker Street (which I can only assume is not cheap) one which a part time Doctor has trouble keeping up the rent for.
In addition he always seems to have scientific equipment available at home when he requires it.
Where does Sherlock's money come from?
In the TV Series, Sherlock calls himself a Consulting Detective. Whilst it is not completely certain that this involves payment, it is certainly conventional for you to pay a consultant in any sphere. Watson is also involved, and spends some of his time blogging about Shelock's successes, presumably not just to enhance his reputation but to 'improve sales'
It is true that Sherlock cannot be bought, he will not take a boring well paid assignment for example - but there is a specific example where he is paid. In The Blind Banker, Sherlock's old university friend who is in a senior position at the Bank offers the two of them a significant fee - Sherlock walks out disinterested in such petty matters but Watson, the more practical one, collects the cheque.
Finally, it is implied that Sherlock is from a wealthy family. It may be the case that he has inherited money. EDIT: Actually from later episodes in Season 3 it seems that while his family are not poor, they don't seem to be ridiculously wealthy - so this can be discounted.
In terms of scientific equipment - Sherlock is probably not above 'blagging' equipment off people, borrowing it for an extended period to help his work.
There is no direct mention of how Holmes receives income, in the television show. However the show draws significantly from the short stories of A.C. Doyle. I am inclined to think that there wouldn't be much difference in the financial life between the two versions of Holmes. Of course Moffat and Gatiss may, in the future, give further insight into their modern Sherlock.
I did turn up this blog: accounting for sherlock which (over)analyses the television show & attempts to shine light on your question. I hope this information helps!
A.C. Doyle did reference how this original Victorian Sherlock Holmes received remuneration. Here is an excerpt from Wikipedia:
… in "The Final Problem", Holmes states that his services to the government of France and the royal house of Scandinavia had left him with enough money to retire comfortably, while in "The Adventure of Black Peter", Watson notes that Holmes would refuse to help the wealthy and powerful if their cases did not interest him, while he could devote weeks at a time to the cases of the most humble clients. Holmes also tells Watson, in "A Case of Identity", of a golden snuff box received from the King of Bohemia after "A Scandal in Bohemia" and a fabulous ring from the Dutch royal family; in "The Adventure of the Bruce-Partington Plans", Holmes receives an emerald tie-pin from Queen Victoria. Other mementos of Holmes's cases are a gold sovereign from Irene Adler ("A Scandal in Bohemia") and an autographed letter of thanks from the French President and a Legion of Honour for tracking down an assassin named Huret ("The Adventure of the Golden Pince-Nez"). In "The Adventure of the Priory School", Holmes rubs his hands with glee when the Duke of Holdernesse notes the 6,000 pound sterling sum, which surprises even Watson, and then pats the cheque, saying, "I am a poor man", an incident that could be dismissed as representative of Holmes's tendency toward sarcastic humour.
I don't think this has been clearly answered in the show, and it has been a long time since I've watched the show. So if there are any errors in my answer it's my memory's fault.
Sherlock's brother in the BBC series is very wealthy and well connected. It's implied that it's his brother who gets Sherlock work as a consultant because Sherlock is too stubborn to find work for himself.
There are a couple of scenes in the show where Sherlock and his brother are at odds with each other, but it appears that Sherlock is somehow dependent upon his brother.
At one point in the show his brother mentions something like "If you applied your intellect to business you could earn a good living". Implying that Sherlock doesn't earn any money, but I always took that scene as an indication his brother supports him financially. So I think it's his brother who receives payments for Sherlock's consulting efforts, and in turn pays for Sherlock's living expenses. I don't think this is the source of his wealth, but more of a brotherly burden.
In the episodes where Sherlock helps the Queen. His brother is the one who arranges for Sherlock to go to Buckingham Palace, and it appears that his brother is motivated financially and has to convince Sherlock to take the job.
There are a few scenes in the show where Watson is left with the tab and has to pay for things like the news paper or a restaurant bill.
Money to Sherlock is either unimportant or other people's problems. So it seems clear that he's not directly responsible for paying the rent bills.
Sherlock is canonically in all iterations a consulting detective, which is pretty much a private detective that helps the police. He also helps a lot of people, so he likely gets discounts or free things, like with Mrs. Hudson and Angelo. Much of his money comes from his clients, but he cares very little for money, and when he forgets Mycroft likely slips some into his funds.
When Sherlock Holmes first met Dr. Watson he had just invented a reagent for hemoglobin. I've always assumed he lived on the patent royalties of that.
