The bike I have came with a 5-speed freewheel. I have seen that there is a single speed freewheel available online. Would that do the trick?

Probably not.

Single speed bikes are picky about chainline so you'd have to check that out.

If you have vertical dropouts, you have to have a way to adjust your chain tension. The simplest way would be to keep your rear derailleur but, if you do that, what's the point?

It will give you one speed, but just using one of the cogs on your five speed will accomplish the same thing. It just won't look cool. The hard part is getting a straight chainline. Read Sheldon Brown.

I've had good luck in using a Shimano BMX freewheel and leaving the rest alone. Maybe I'm lucky--chainline be damned!

Me too. There are a few cheap things you can do to move the front or rear one way or the other though. On the rear, screw a bottom bracket nut (same threads, free from your LBS) on before your BMX freewheel, thereby moving it out a few mm. On the front, if you have a double crank, you can put the chainring on the inside or outside of the crank.

Vertical dropouts can actually work better for single speeds if you can find the gear combination that will give you correct chain tension. The wheel is locked into position, so it can't be pulled out of position no matter how hard you stomp on the pedals. You can even use a quick release. When the chain gets too loose, it's telling you that it's worn out.

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