A brief survey of what some of the world's subways are like:
London: It looks dilapidated. It smells dilapidated. It is dilapidated. But comparitively, it's not that bad! The stations are plentiful and modern. The grid is layed out as well as could be expected. Getting around with them is very easy.

Amsterdam: They have a subway system. Really. I think it has only one line or something. It is very inconvenient, and is very warm. The cars are kind of nasty, but the station isn't overly bad. Riding the subway in Amsterdam is far more trouble than it's worth. The trams are much nicer and easier to get around on.

Paris: The Paris subway is really bad. The grid is crazy--there is a glut stations in places where there is no need for a glut of stations, and there are no stations in places where there really should be. Getting from point A to point B is really difficult because of the insane way it is laid out (or maybe it just seemed that way because Zeno was right. The stations are really old and ugly and hard to get around. And warm. Really warm.
London Underground : Extremely hot, clammy and urine-scented in many cases. Trains range from modern and shiny filth pits to dilapidated and flea-riddled filth pits. Mice are present on tracks and platforms. Very crowded at peak times, although not as bad as Tokyo.

Technological achievement : the newest line sports trains so smooth you can hold a cup of coffee without spilling it.

Biological achievement : a new species of mosquito has evolved (over 4,000 generations) that has no light-sensitive organs and can survive on minimal moisture, and can even hibernate for extremely extended periods of time. Was introduced into the tunnel environment during the Blitz.

Financial achievement : Travelling on the tube is more expensive per mile than Concorde.

Tip for new users : Although there is a £10 fine for failing to carry a valid ticket when attempting to exit a station, you can bribe the desk staff quite easily. I have successfully done so at Seven Sisters station. (Usually I carry the correct ticket, but I somehow managed to cross zone lines).
Toronto Subway: Began in mid-50's, expanded in 60's, 70's, and 80's. Originally envisioned as the solution to transportation problems in the city. Hopes foundered on arguments between leading population growth, and following. No longer believed to be adequate for the need, but better than nothing. With the end of provincial subsidies, thanks to Mike Harris, much more expensive.

(The original subway from Eglington to Union Station was mostly trench built, as Whywait? writes. The first new leg, from Union Station to St. George, was mostly tunnelled. You can see this in the shape of the underground itself: St. Andrew, St. Patrick Stations are rounded. It was tunnelled, I believe, because it was much too much trouble to make tunnelles in the business district, and near The Pink Palace. The Spadina and further north subway, is mostly built above ground as I remember.)

Montreal Metro: Unique in that it travels on rubber wheels. Is without the tooth-wrenching screeching all metal-wheeled subways do. Built to reach the World Exposition held in 1967, it joins islands and suburbs.

Hong Kong Very shiny, clean and efficient subway system. It's remarkably easy to slide off the metal seats if you're not paying attention to corners.

Warsaw has a single line subway system. Bugger all use. The city council chose to have a Palace of Culture from Moscow instead when offerered a choice between that and a decent metro system.

London Underground is hellish. Useful, but foul. One of the oldest, one of the most run-down and underfunded in the world. But it's still my favourite zoo.

Paris is a confusing subway system, but, at least the old art nouveau station entrances are pretty. The tunnels are spooky.

the New York Subway is filled with stories.
Glasgow has its own underground rail network, with two lines and 15 stations serving the centre, west and south of the city. The trains are tiny, cute, and orange. Don't tell Glaswegians that it's just like a miniature London Underground because by and large they don't like London.

Singapore has the MRT which like most subways has underground and outdoor lengths of track. It is very clean, with downtown stations equipped with the newest in anti-suicide automatic sliding glass doors to prevent "person(s) at track level". It is also true that Lee Kwan Yew, Sinkers' "benevolent dictator", banned chewing gum from the city-state after hoodlums and lepakers discovered that jamming a wad between the doors shut down the entire system.

Kuala Lumpur began building the STAR LRT in 1993. One line runs to Putrajaya, the city's planned administrative hub; another goes to the site of the 1998 Commonwealth Games. I think there are three lines finished by now. As I was moving away from Kuala Lumpur the British engineering firm Taylor-Woodrow were trying to dig the only two sub-surface stations in the system but the tropical soil (already extensively mined for tin and full of caves and sinkholes) wasn't cooperating and their tunnels kept collapsing. The legoland-ish trains are spiffy, though.

Toronto's "Rocket" and the New York subway are trench-built subways, while the Glasgow and London Undergrounds were tunnelled, sometimes hundreds of feet below the street. Trench-built tunnels run mainly just below street level and are much safer. Tunnelled subways give one the feeling of travelling in the bowels of an entity with a sentience of its own, and are therefore, without question, far superior.
A few additions for places not already covered in previous writeups...

Scandinavia

Finland: The Helsinki Metro is based on designs completed in the 1970s and looks like it. Stations are utilitarian and made from concrete, while the trains are orange boxes. But it's fast, efficient, and tolerably clean. It has exactly one line starting from Ruoholahti, going through the city center, forking at Itäkeskus (Scandinavia's largest shopping mall!) and terminating at Mellunmäki and Vuosaari

Sweden: The Stockholm subway, or Tunnelbana ("Tunnel Track"), consists of three lines constructed between 1950 and 1994. All three lines join at T-Centralen, smack dab in the middle of the city. The green and red lines are pretty utilitarian, but the blue line stations are filled with all sorts of weird and wonderful art and has thus been dubbed "the world's longest art exhibition". Part of the funkiness are the stations themselves, which incorporate the raw cavern walls blasted out of rock instead of just covering them up with concrete. Fansite: http://english.tunnelbana.com/

Norway: The Oslo subway has no less than 5 lines serving all of 500,000 people, making it probably the most over-subwayed city on the planet. The network was constructed essentially by digging one tunnel through the center and piping suburban lines into it, resulting in a rather awkward map.

Denmark: Despite being the largest city in Scandinavia, Copenhagen has no subway network at all, although the suburban train network plays a similar role. This is about to be remedied though, as the first segment of Copenhagen's first subway line will be was opened in October 2002.

Continental Europe

Germany: Berlin has an extensive subway network of no less than 9 lines, dubbed the U-Bahn in German. Divided in two during the Cold War, the two halves have since been patched together and detecting the former border is getting harder day by day. As you might expect, everything works with German efficiency.

Italy: Rome has two subway lines and a third under construction. The system has a pretty bad rep for overcrowding, dirtiness and thieves, although I didn't find it particularly bad. In an effort to up ridership, two years ago McDonalds was running a campaign to give discounted hamburgers to anybody who brought in a used metro ticket.

Hungary: Budapest's first metro line (földalatti) was completed in 1896, making it the first subway in continental Europe. (It was recently restored to its former glory for its centennial.) Two more lines were added by the Communists, but the construction of the sorely-needed and long-awaited M4 line from Keleti to southern Buda remained embroiled in a quagmire of lawsuits and financial problems.

Russia: St. Petersburg, formerly known as Leningrad, has a Soviet-designed subway system that is many ways quite an achievement. As the entire city is built on a bog, the tunnels had to be dug quite deep, and having to ford a number of rivers and canals along the way didn't make life easier. There are currently 4 lines and a fifth under construction. A few odd details: all stations have doors between the platforms and the trains (not to prevent suicides, but to allow time to escape if the tunnel roof springs a leak!), and the displays count time up from the previous train departure. Intervals may be as little as 30 seconds at rush hour.

Russia: Moscow has more of the same, only bigger and better with chandeliers in some of the main stations and no less than 11 lines in operation.

Africa

Egypt: The Cairo subway is the only one on the entire African continent. Line 1 from Helwan to El-Marg used the Oslo approach by building a tunnel between two commuter trains and calling the result a subway. Line 2, from Shobra to Giza (read: the Pyramids), was built from scratch, and a line 3 is and probably will remain on the drawing board. The three intersections of the lines are dubbed, predictably enough, Mubarak, Sadat and Nasser. The system is tolerably clean and navigation is a charm compared to most alternatives, but the ticket sellers will happily rip off khawagas who can't decipher the Arabic-only fare tables. Then again, with tickets clocking in at 50-80 piasters ($0.10-0.15), most tourists will never notice the missing 20 pt... although, come to think of it, most tourists will use taxis and never notice that they're being charged triple. Inshallah.

Asia

Turkey: Istanbul -- whose subway system is entirely on the Asian side at time of writing -- has Asia's oldest subway (1875), the 2-station and 500-meter funicular dubbed the Tünel, which climbs up a hillside from Karaköy to Galata. Partly open but still under construction is Istanbul's first "real" metro, which currently stretches from Levent to Taksim (the center of modern Istanbul) and is being extended across the Golden Horn to the European shore and the old city.

Japan: The Tokyo subway system is nothing short of amazing on any scale, see sekicho's writeup for details. One telling anecdote: during 5 years of residence in the city, I've had to use a bus once, and I later realized that the subway station recently constructed in the vicinity was just missing from my old map. Also note that, in those same 5 years, I have never seen or felt a pusher: they do exist, but are only around if you're at the wrong station going on the wrong line in the wrong direction at the wrong time. (Shinjuku, Marunouchi, south and 8 AM should do the trick.)

References and Fun Browsing

http://www.metropla.net not only has a punny URL but gobs of info on every single subway, monorail and LRT on the planet. Check it out!

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