What unusual means of public/share transportation exist around the world?

In the Netherlands, on the same base as hitch hiking, hop on the back of a bicycle.
Do wait to be invited, but if you try the ‘hitch a ride thumb’ next to a busy cycle track you might be surprised by several offers in a very short time.

Hitching a ride on a bike, the Dutch way. By João Pimentel Ferreira See the attributions here.

This youtube video shows you how to do it, been around for a few years now and rather popular.
In short, run a few steps next to the bike to pick up speed, grab the rider and hop on, a little hop to adjust your seat and cross your ankles.

I met two girls while in France, one had been in the Netherlands, the other going there. The tip was, accept a ride on a bike once. Once is enough though.

The Tateyama Tunnel Trolley Bus in Toyama, Japan, is not such an unusual vehicle; however the route, which is 3.7km long and located completely underground, may qualify it for inclusion here.

On the other side of the valley, the Kanden Tunnel Trolley Bus goes for 6.1km, but surfaces into fresh air slightly before its terminus. Together, these are the only two trolley buses in Japan.

How about a tanga (tonga) – which is a shared ride on a horse drawn carriage, indigenous to India, Pakistan and Bangladesh:

(image credit: wikipedia)

Note this is not like the ornate horse drawn carriage rides in parks and similar; this is a daily means of transport for thousands of people – I have taken it many times to go to the market for groceries, etc.

The cable cars in Chiatura, Georgia were built in the 1950’s to transport mineworkers between the city and the mines in a fast way.
Now they’re still in use, not sure about the price and capacity of these cars though.
Here’s a photo report of the cable cars in Chiatura.

In Santorini, Greece there are mules or donkeys and cable cars to cope with the steepness of the volcano shaped island.

Brunei.
In Bandar Seri Begawan – the only town/city of any size, a significant proportion of the people live in the ‘Kampong Ayer” = Water Village – a widespread collection of homes on stilts and platforms spread along both sides of an inlet / river. Some of the buildings are very ‘flash’, and while some are far more modest, it is not anywhere a slum as might be seen in other countries. A useful amount of the country’s oil wealth manages to ‘trickle down’ to the lower levels of society so that there are essentially no ‘desperately poor’ people.

To provide access to & from Kampong Ayer, long thin motorboats with large outboard motors are used to transport all and sundry, usually at breakneck speed, to and from the village at all times of the day or night.

Patrons can be veiled women shrouded in Islamic dress, children, shoppers or businessmen.

Many images here

Kampong Ayer

• one view of many many many …

Source: Wikipedia

“Taxi !!!…”
Source: http://shelf3d.com/i/Bandar%20Seri%20Begawan

Besides tourist-focused gondolas, Venice offers canal crossing using public Traghetto.

(Picture from Wikipedia)

Not the class of answer you probably expected, but it seems to meet your criteria, more or less*.

In Shanghai, China, you can catch a “train” from Pudong International Airport.
It’s the only one of its sort in China and not too common worldwide.
Was and may still be the fastest commercial train service in the world.
Peak speed is 430 kph, making it the fastest I’ve ever travelled on land.
It’s a Maglev train – no wheels, no ground contact, feels like flying, looks like flying (IS flying). German ‘Transrapid’ design. They hoped to sell them to China for use elsewhere but this was the first and the last – China instead adopted the convention “D Train” which runs at closer to 200 kph.)

Wikipedia

Above photo is cropped from image on above Wikipedia page.

Over some of the route the Maglev runs alongside a motorway with a max speed limit for cars of 100 kph. At around 400 kph the train passes cars as if they are standing still.
There are two trains on the track. They pass halfway along the route on a double parallel section. At an effective approach speed of around 800 kph the passing is impressive.
Bang, blurred impression of motion outside windows, chucketa chucketa chucketa , … gone. A few seconds. 100 metres of train. 0.5 seconds. Awesome.

5 minute video here (not mine) – passing happens at about 3m-06s. Takes around 1 second for 2 x 100m trains. [800 kph ~= 220 m/s so for your window to traverse whole 100m of other trains takes about 0.5 seconds. From first crossing to ends passing takes about 1 second.

Fare is not cheap compared to bus or train to town, but is reasonable given the uniqueness and experience. Without looking it up, maybe $US10. It’s fast BUT stops on the edge of the city – near an Expo centre that it served when first opened. You can catch bus or taxi from there to city centre but it adds inconvenience. There is also a nearby subway which is quite a good way to get to and from it.

• Maglev, only in Shanghai in China, possibly fastest commercially operating in world.

In southern Saudi Arabia in an area on the top of few mountains which has a very unique and complicated terrain, people need a unique way of traveling between villages of that area. This is how (Fifa’a) looks like:

And this is how they travel from one village to another, or from one mountain to another:

These are local made cable cars, I have no Idea how do they make it.. but they do. I wouldn’t dare to be inside one of those.

BTW, Until around 100 years ago or so this area was totally isolated due to its nature. People lived there without going to the outside world for hundred of years. That’s why this is the only area in Saudi Arabia which has its own language and not Arabic.

The Cambodian “bamboo train” is a nifty solution to a once-extensive railway that fell into disrepair during the country’s lengthy civil wars. Take two pairs of wheels, plop them on a bamboo frame, tack on a dinky little engine and voila:

(courtesy Noud W on flickr, CC by-nd)

So what do you do when two of them meet? Dismantle the one carrying less cargo, carry it past the other, set it up again and off you go!

Here’s a 2006 BBC article on them. There are probably less of these than there used to be now that Cambodia’s actually started refurbishing the network though.

I’m not sure whether you are looking for amusing unusual or just plain unusual. But here it goes anyway.

This is the SkyLab. (Yes, we really call it skylab and you will see why.)

This is basically a motorcycle with a plank of wood balanced across. And if you are wondering why on earth would someone do such a thing outside of X-Games, the answer is poverty tends to make people creative but couple that with a very lowered sense of personal safety and you get the SkyLab™. You will find these at some rural places in the Philippines. There are also some videos in YouTube for this.

P.S. It is called skylab because it looks like the space station from behind. Just picture the horizontal plank as the solar panels of the SkyLab. Here’s an image from flickr of the skylab from behind. Note the improvements made on passenger comfort 🙂 http://www.flickr.com/photos/24265223@N04/2743621223/

The Schwebebahn in Wuppertal, Germany is a suspension railway first opened in 1901 and still in operation as a regular mass transit system.

Each train can carry around 120 passengers, and a one-way ticket is 2.40 EUR. During rush hours there is one train every 3-4 minutes.

There are a handful other (much newer) suspension railway lines in operation in other places, e.g. Memphis and Tokyo.