There were 1976 filings for the name itself as a logo:
These are no longer active. Then there were filings for a new version, in 1984:
The series ended in 1989 but oddly there was a 1996 filing, so presumably money was still being made from fans:
There was the filings in 2005, when the show was relaunched.
In 2009 the last look, in versions including a black and white look:
There were more than one application for each as different services and product areas were involved. Each logo reflects a different cultural look, suitable for its time, and the last two certainly reflect the expensive look of the modern show.
The official database I used to find these trade marks doesn't provide a link to lists of results (why not ?), so to find the 28 registrations listed for DOCTOR WHO go to the official website and enter that wording as a Word search, search type "contains string", The results, if clicked on, show the services or products involved for each registration. Some are EU registrations, available from the newish OHIM office.
Turning to the means by which the Doctor travels, there's the image of the Tardis in UK trade mark 2104259:
It was applied for in 1996, and was the subject of a dispute with the Metropolitan Police, as it is a police telephone box, where members of the public telephone the police for help. It was registered in 2002 after a Patent Office examiner in a hearing pointed out that the police boxes weren't used for that anymore. The decision gives 15 pages discussing the matter, and rejecting the Metropolitan Police's objection.
THE TARDIS, meanwhile, was safely registered in 1976, as was, oddly enough, a registration for that word plus a black and white image of a police box, also in 1976, as UK trade mark 1068700, though only for toys and games, as shown below.
THE DALEKS was registered as long ago as 1964, while CYBERMEN dates back to 1996 (these are enemies of the good Doctor). Even SONIC SCREWDRIVER has been registered (sadly, not for tools).
I see that two 2012 applications through the European system by Canal + Image UK, DR WHO AND THE DALEKS and DALEKS' INVASION EARTH: 2150 AD have been opposed by the BBC and hence are awaiting a decision.
Happy anniversary, Doctor Who.