Part of the role of the OpenStreetMap Foundation is to hold and manage OSM related intellectual property (trade and service marks, domain names) for the community.
The OpenStreetMap Foundation owns the EU community trademark for "OpenStreetMap", and for the magnifying glass logo, a UK trademark for the logo and the US trademark for "OpenStreetMap". Registration in other areas is in process.
If you have questions about acceptable use of the trademarks or related questions, please contact the LWG (firstname.lastname@example.org).
OpenStreetMap logo application (UK)
Application number UK00002500155 for the OpenStreetMap logo was filed at the UK Intellectual Property Office on 17 October 08. The original applicant was Steve Coast.
It passed examination and was published on the 21st November.
There was subsequently an agreement by Steve Coast at the November 2008 board meeting that it should be transferred to the Foundation. It was also stated at the November board meeting that the application was likely to fail.
The application was still in Steve Coast's personal name as of 19th Jan 2009 when Peter Miller lodged an 'Intention to Object' to provide an addition month for the application to be transfer to the foundation as agreed.
On the 19 January 2009 it was reported that the OSMF Secretary was working with a trademark law firm to get the legal ownership records updated with an update expected 'by the end of the week'.
The trademark application was approved by the IPO on 27 February 2009.
Steve Coast was still shown as the proprietor of the trademark on 07 July 2009.
The trademark assignment was signed by Steve Coast at the State Of The Map 2009 conference on 12 July 2009 and as of 15 July 2009 the trademark as been shown by the IPO as belonging to Openstreetmap Foundation.
OpenStreetMap name and logo application (EU)
Application number EU007366859  for the OpenStreetMap name (in English and Spanish) was filed with the EU Intellectual Property Office on the 3rd November 2008. The application was made in Steve Coast's personal name.
It was agreed by Steve Coast at the November 08 board meeting that it should be transferred to the Foundation. It was also stated at the November board meeting that the application was likely to fail.
The trademark application was handed over to the OSM Foundation secretary during December 2008.
On 19 January 2009 the application was in Steve Coast's personal name. On inquiry is was reported that the OSMF Secretary was working with a trademark law firm to get the legal ownership records updated with an update expected 'by the end of the week'.
On 07 July 2009 Steve Coast was still shown as the applicant for the trademark on the EU Intellectual Property Office's records.
The trademark registration was finally transferred to the OSMF on May the 28th 2013, as was the EU registration of the logo (CTM 007366991).
OpenStreetMap name application (UK)
Application number 250154 for the OpenStreetMap name was filed by Steve Coast at the UK Intellectual Property Office on the 15th October in his own name and has been examined.
There was subsequently an agreement by Steve Coast at the November 08 board meeting that ownership of the application should be transferred to the Foundation. It was also stated at the November board meeting that the application had failed.
The trademark application process was 'handed over to the OSM Foundation secretary' during December 2008.
On the 19 January 2009 records showed that the application was in Steve Coast's name.
On the 07 July 2009 Steve Coast was shown as the proprietor on the IPO web site.
On 19 January 2009 it was reported that the OSMF Secretary was working with a trademark law firm to get the legal ownership records updated with an update expected 'by the end of the week'.
On 8 July 2009 the trademark was shown as 'refused' on the IPO website with Steve Coast as applicant. It is understood that it failed because it was considered to be a descriptive name for the project and as such could not protected in this way.
Concern was expressed in late 2008 within the OpenStreetMap community when it emerged that Steve Coast was the personal applicant for the OpenStreetMap trademark. It then emerged that he also still owned some key domain names and was handling discussions in relation to the Open Data License on behalf of the Foundation with the same lawyers who had advised the commercial company in the 'hostile takeover' of Twiki . The issue was described as 'a storm in a tea-cup', as 'a mistake'. Steve and the board reassured the community that the trademarks and domain names were in the process of being transferred promptly. Control of domains and Trademarks have subsequently all been handed over to Foundation. EU trademark legal paperwork is ongoing (as of August 2009), but it is already owned by the Foundation.
Note: the above statements wrt the transfer of registration do not correspond to the actual facts, in particular there had been no documented attempt to actually transfer the trademark before early 2013.
In April 2013 the CTM trademark was still owned by Steve Coast not the OpenStreetMap Foundation. 
The trademark registration was finally transferred to the OSMF on May the 28th 2013.
OpenStreetMap US Trademark
Registration 4566302 July 15th 2014 https://tsdr.uspto.gov/#caseNumber=4566302&caseSearchType=US_APPLICATION&caseType=DEFAULT&searchType=statusSearch
Expansion of OpenStreetMap Mark Coverage
Registration of the OpenStreetMap mark via the Madrid protocol has been applied for in Australia, China, India, Philippines, Russian Federation and Ukraine.
For coverage of the CTM and the process of the MP see http://www.inta.org/TrademarkBasics/FactSheets/Pages/CTMMadridComparisonFactSheet.aspx
Protection in Australia was granted on May 13th 2015: reg. number 1646247
Protection in the Philippines was granted on April 20th 2015: doc. number 2015/93389
Protection in Ukraine was granted on September 2nd, 2015
Protection in Turkey was granted on November 16th, 2018
Protection in Brazil was granted on May 21st 2019, 914586726 & 914586890
Protection in Argentina was granted on September27th, 2019, 3025089
OpenStreetMap Japanese Trademark
The OpenStreetMap trademark is currently owned by the Japanese OSM association.
"State of the Map" Trademark
EU (CTM) protection granted May 27th 2016 Registration number 015029911
US protection granted October 16th, 2018, Reg. 5583071 United States Patent and Trademark Office website search results
OSMF Trademark Policy
The Trademark Policy is currently under discussion, however you should assume that the final version is unlikely to be very different.