Europe

Below is an incomplete list of domestic public transportation planners. Those usually do not include international transportation, except trains in the case of Bahn.de. Intercity buses may or may not be included, depending on the country.

Austria

• oebb.at. Similar to the German/Swiss example, although this is actually a railway company, it includes bus-stops. I don't know if it includes all public transportation.
• For Vienna: Check out the homepage of the "Wiener Linien" and the mobile app WienMobil.

Finland

• Journey.fi Rail and bus connections, flights, and walking routes in whole Finland; also local transport connections for 21 cities.
• Matkahuolto The official resource for buses. Includes the very nice option to not specify a date, and thus find out connections that run only on specific weekdays.
• VR The official resource for all train connections.
• Many cities have their own local transport journey planners: Helsinki region (Reittiopas.fi), Tampere, Turku (Föli), Oulu, ...

France

• Voyages-sncf - for trains only, nation-wide. They launch a new feature, quite interesting, for travel by train/plane/car across Europe (it compares them, it does not combine them). For urban transportation, every city has its own system. For Paris region, ViaNavigo is a helpful search engine.
• For Paris - ratp.fr RATP is the public transportation company operating in the Paris greater metropolitan area.

Germany

• Bahn.de. Although Bahn is a railway company, its search engine includes at least buses as well. For example, here is a search from Strausberger Platz, Berlin to Vallun Chafuol, Zernez — from a metro station to a bus stop in Switzerland. I don't know if it includes all public transportation.
• Busliniensuche.de. It's a search portal that currently offers a search function across 57 companies running one or more bus lines.
• Busticket.de - another search portal for buses.

Italy

• trenitalia.com - Trenitalia is the Government-owned rail transportation company in Italy
• italotreno.it - Italo is a railway service provided by Nuovo Trasporto Viaggiatory. This is a private-owned company founded in 2006 to provide some form of competition to the former Trenitalia monopoly. This helped bring the overall prices down significantly, over the past few years.
• For Rome - atac.roma.it ATAC is the public transportation company operating in the Rome greater metropolitan area
• For Milan - atm.it ATM is the public transportation company operating in the Milan greater metropolitan area

Liechtenstein

• lba.li In Liechtenstein there are no trains, but this site offers an queryable online schedule of the public bus system. This information is also available at sbb.ch.

Netherlands

• 9292ov.nl. Includes all public transportation in the Netherlands and combines train, tram, bus and metro in a single itinerary.
• ns.nl. Biggest train company in the Netherlands, also includes information from other railroad companies. This website also shows up-to-date information about delays.

Norway

• rutebok.no. Unlike most domestic planners, this includes flights.
• ruter.no the best for Oslo and surrounding area.

Poland

• jakdojade.pl - public transport in main Polish cities
• e-Podró?nik - search engine for bus connections between cities. Shows also train connections.
• BlaBlaCar - search engine for bus connections between cities both domestic and international. You can buy bus ticket via this site.

Portugal

• www.cp.pt Trains all over the country except for the line crossing from Lisbon to Setubal which is operated by another company named Fertagus
• Transporlis All transports in Lisbon metropolitan area

Russia

• rzd.ru. Only trains. Contains quite some trains not listed in HAFAS and also quite some international connections, perhaps also trains in several former Soviet republics — I don't know.
• eng.rzd.ru, modern English language version for Russian trains including international connections.

Spain

• Renfe has a timetable search for trains only, and quite poorly implemented (e.g. at most one change included). From my own experience, Bahn.de and other HAFAS-based sites are often incomplete in Spain. There are many different local companies operating buses; some are included in Google Transit, but some are not; if you're going to a specific city, just try.

Sweden

• Resrobot. In my experience, it does not have all local buses within every town.
• For Stockholm region: reseplanerare.sl.se

Switzerland

• sbb.ch. Provides schedules for train, bus, and ships all over Switzerland. Also each city is included.

UK

• Traveline has a number of regional journey planners, together covering all of England, Scotland and Wales. Including buses, trains, ferries and flights.
• For London: Transport for London Journey Planner
• For Northern Ireland: Translink Journey Planner
• For Manchester: Bus Journey Planner for Manchester

Fallback

• For most trains in most of Europe, any HAFAS-based search engine such as Bahn.de, B-Rail, or several others. Note that HAFAS is not complete. See also this post on the risks with HAFAS.
• For buses or cities in countries or destinations not listed above, maybe Google Transit works, but more often than not...
• Try googling: journey planner <city> or journey planner <country>

North America

Google Transit offers global transit planning, with very thorough coverage of the U.S. and Canada— in fact, most of the participating agencies use Google in lieu of an independent service. It is accessible in Google Maps.

HopStop similarly provides trip planning for nearly all major and many minor cities in the U.S. and Canada, and offers iOS, Android, and Windows Phone apps.

Canada

• ViaRail for the train connections in the corridor between Quebec and Vancouver.
• Greyhound for most bus connections (inter-city) west of Montreal, Orleans Express for some of Quebec and Maritime Bus for Atlantic provinces. Intercar serves the North-East of Quebec and North coast of St-Lawrence, Autobus Maheux the North of Quebec and multiple other carriers serve parts of Quebec province.

Local

• Montreal - STM Trip Planner
• Ottawa - OC Transpo Quick Planner
• Toronto - TTC Trip Planner
• Vancouver - Translink Trip Planner

Mexico

• TicketBus.com.mx for intercity schedules across a variety of bus companies.

• Mexico City (Distrito Federale) - Metro Origin-Destination (English)

Puerto Rico

• San Juan - Tren Urbano Itinerario (Spanish)

USA

• Amtrak for intercity trains in the USA.
• Greyhound for most intercity bus connections.

Local

Most local and regional transit planners are provided by the operating agencies, and may not reflect all options. For example, commuter rail and rapid transit are often administered separately, and a trip planner for one service will often exclude other options.

• Baltimore - MTA Trip Planner
• Boston - MBTA Trip Planner
• Buffalo - Metro Trip Planner
• Connecticut - use Google Transit
• New Jersey - NJ Transit Trip Planner
• New York City - MTA Trip Planner
• Philadelphia - SEPTA Trip Planner
• Pittsburgh - Port Authority Trip Planner
• Rhode Island - use Google Transit
• Washington, D.C. - WMATA Trip Planner

• Atlanta - MARTA MyCommute
• Austin - CapMetro Trip Planner
• Charlotte - CATS Trip Planner
• Dallas - DART Trip Plan
• El Paso - use Google Transit
• Fort Worth - use Google Transit
• Houston - Metro Trip Planning
• Jacksonville - JTA Trip Plan
• Louisville - use Google Transit
• Memphis - use Google Transit
• Miami - Miami-Dade Transit Trip Planner
• Nashville - use Google Transit
• New Orleans - use Google Transit
• Orlando - use Google Transit
• Raleigh/Durham - use Google Transit
• San Antonio - VIA Trip Planner
• Tampa Bay - use Google Transit

• Chicago - Goroo
• Cincinnati - use Google Transit
• Cleveland - use Google Transit
• Columbus - use Google Transit
• Detroit - use Google Transit
• Indianapolis - use Google Transit
• Kansas City - use Google Transit
• Minneapolis/St. Paul - Metro Transit Trip Planner
• St. Louis - use Google Transit
• Tulsa - MILES

• Denver - RTD Trip Planner
• Honolulu - use Google Transit
• Las Vegas - use Google Transit
• Los Angeles (inc. Orange County and Ventura County) - go511.com
• Phoenix - ValleyM Metro Trip Planner
• Portland - TriMet Trip Planner
• Sacramento - Regional Transit Trip Planning
• Salt Lake City - UTA Trip Planner
• San Diego - OTIS Trip Planner
• San Francisco Bay Area - 511.org
• Seattle-Tacoma - Community Transit Trip Planner
• Spokane - use Google Transit

Is there a list of such services in other countries somewhere?

No, I don't think such a global list exists—at least anything that's comprehensive and up-to-date.

Instead, your best bet is to simply google:

journey planner <city> 

This works for pretty much every city that has an online journey planner (also called route planner or trip planner). Including London.

Update: The answers to this question right here might be the best attempt at such list. :-) Still, they are far from comprehensive (there are many cities with a journey planner).

Australia

Sydney

• NSW TransportInfo "Plan Your Trip"
  has more thorough options than Google Maps for Sydney.

Melbourne

• Public Transport Victoria's Journey Planner

You should try Google Maps. There you can enter a starting point and a destination, and Google Maps calculates routes for you. In most countries. respectively regions, public transport, private motorized traffic, and pedestrian routes are covered.

For example if you want to go from the Buckingham Palace to the Hardrock Cafe in the neighbourhood, Google Maps gives you the following route if you want to travel by car:

But if you're keen to walk, you get another route:

Many public transport systems will have similar planners on their website, so that's a good place to start. eg, if you're in New York, the MTA website has a good route planner. For London, it's on the Transport for London website.

The other option that provides coverage for multiple cities in multiple countries is Google Maps which allows you to search for a trip via car, public transport, walking or bicycle - although the exact options will vary depending on the region/city. Google also has the ability to start or end a trip at "My Location" (just type that into the location box) and it will do it's best to automatically work out exactly where you are.

The Middle East

Israel

*-not including Palestinian territories in the West Bank and the Gaza strip.

trains

The train system in Israel is under one national company, so all train times, destinations and prices can be viewed on the Israel Railways site.

Train tickets are bought at the station, either from a cashier or an automatic machine.

buses

There are several bus companies in Israel, most of them operate in different areas. This site allows to search for buses from all companies, and has links to all the sites of the different companies. The site gives both lines inside cities and between cities. Includes buses to territories under Israeli control in the west bank.

Tickets are usually bought from the driver in the bus. If you get to the bus in a central station, a ticket can be bought in the station, but isn't mandatory.

The only exception is buses to Eilat. Tickets to Eilat and its surroundings need to be bought in advance on-line or at a central bus station. Site for Eilat booking

other

Inside Jerusalem exists a light rail: their site.

An underground train exists only in Haifa, and it has only six stations, the Carmelit's site

Flights are available to/from Haifa, Tel-Aviv and Eilat. Tickets are bought trough the airline site, or through travel agencies or flight booking sites. Sites for the airliners: El-Al, Arkia and Israir.

To answer your query about transport planning for London, "Transport for London" (TfL) has an A to B journey planner available on their website:

http://journeyplanner.tfl.gov.uk/

You can set numerous options including date/time of travel, preferred mode of transport, mobility needs (impaired mobility?) and whether you prefer speed vs convenience (changes/walking distance).

Note that A and B can be a station, bus stop, postcode (known as zip code in some locations) or 'place of interest'.

In addition for Australia:

Western Australia Transperth quite competently covers most of the state for bus, rail and water public transport.

http://www.transperth.wa.gov

Victoria Public Transport Victoria- never used it but should cover Victoria
http://ptv.vic.gov.au

Queensland Translink- had to tell the coverage but at least covers the major tourist areas of Cairns and Gold Coast.

http://translink.com.au/

Japan

https://world.jorudan.co.jp/mln/en/?sub_lang=nosub

(Never tried this English version, but have used the Japanese version for ca 20 years, works well, and at least nowadays also includes domestic flights) (Often cheaper than train)