Is it possible to convert a single speed rear wheel into a multi-speed rear wheel?

No. Not in any practical sense.

Single speed hubs cannot be converted to multi speed. You'd have to swap the hub out in which case you might as well buy a multispeed wheel in the first place.

Conventional wire spoked wheels don't allow this. All types of conventional singlespeed hub (BMX, cassette, freewheel, track, coaster) take advantage of their one-cog-ness to have center-to-flange measurements that are either symmetrical or close to it, creating a wheel with no or almost no spoke tension disparity, which makes it stronger and more robust. (Non-flip-flop track and all disc singlespeed hubs do give some spoke tension disparity, but not much.) Such a hub offers no space to put a gear cluster.

Certain disc (as in flat and expensive) and mag wheelsets are switchable from track to multi-speed hub guts. They're not subject to the above limitations as their structure is closer to only existing in one plane. To do it you're buying the manufacturer's parts and following their procedures. Zipp 900s are a current example, allowing riders to use the same big money disc for both track and TT/tri. Aerospoke also is/was switchable, and were pretty common in their heyday, so probably represent the most popular singlespeed wheelset you can buy and then convert later to derailer use.

You want to use a track wheelset as a "multispeed rear wheel"

First problem is the Over Locknut Dimension/Distance of the wheels and the spacing of the frame.

A track wheel will probably have an OLD of 120mm, the horizontal distance between the outsides of the two locknuts. This should be the same as the space between the rear dropouts on your frame.

If your frame was for a multispeed bike, it could be 126mm, 130mm, 135mm, or a lot more. This distance scales approximately with age. A 130mm OLD could have been an 8/9/10 speed, and a 126mm might have been a 5/6/7 speed cassette.

If the OLD doesn't match you might be able to compress / bend / cold set the frame to fit. But squashing one up evenly is a lot harder than expanding an old frame. Frame must be steel to do this - an aluminium or carbon or titanium bike will not cold-set.

You might also get away with adding some spacers to the outside of the wheel's axle, but it will be inherently weaker because of the unsupported length of axle. I did this to get a 130mm wheel in a 135mm frame temporarily, and the axle bent within a month.

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Another option is to get multiple speeds by running an internally geared hub. Downside is this is significantly expensive and requires a full rear wheel rebuild.

A Rohloff starts at 135mm wide, but offers 14 gears.

If your frame is 120mm OLD (ie its a track frame) then you have some options but fewer gears. These are 3-5 gears.

• Sturmey Archer SRF3 (118.9 mm OLD)
• SRAM Spectro T3 (117 mm OLD)
• Sturmey Archer XRF5 (119.7 mm OLD)

From http://www.hubgear.net/table.html A good resource there.

Regardless of your chosen solution, there is no cheap/affordable way to do this well. If you really want gears, buy a bike with gears and keep your existing one for whatever riding suits it.

Knutson is of course technically correct.

However, I have successfully used a track wheel to mount a 6 speed freewheel (which has the same attachment spec as a single) by replacing the wheel axle for one of the correct length for the frame and dimensions for the hub, spacing it appropriately and dishing the wheel. It's straightforward but best reserved for narrower rear triangles due to the limitations of the freewheel design plus the amount you'll be able to dish without the spoke length coming into play.

many older road bikes used a flip-flop track hub in this way, so the type of hub is not a limiting factor.