How can I measure cadence without attaching anything to the bike?

This is a somewhat alternative solution, but you can use music. Find a tune that has a similar speed to your cadence, and google for "BPM (song name artist name)"

https://tunebat.com/Info/The-Gambler-Kenny-Rogers/5KqldkCunQ2rWxruMEtGh0

So "The Gambler" by Kenny Rogers is 87 BPM, meaning you're pedalling at 87 RPM, or maybe at ~44 RPM if you're on the half-beat not the quarter beat.

This presumes you can listen to music while riding, that's another topic.

You can flip this around, and if there's a cadence you want to ride at, then find music which has a speed of the same, or double. You might want to practice at 100 RPM and so "Guns and Roses - Paradise City" is 200 BPM. Find close-enough matches with services like https://jog.fm/popular-workout-songs?bpm=200

Oh, measuring cadence is simple: Find a nice silent road where you can ride for a minute undisturbed, take some kind of clock that shows seconds, and then simply start counting when the seconds reach some set number (00 is perfect, but I find that 10, 20, etc. do just as well). Stop counting when that set number is reached again, and the number where you stopped counting is your RPM.

This works for counting heart rate (at rest, not on bike; most people can't easily count that fast) just as well as counting pedal strokes, and is precise to a single beat/stroke. You just need to be able to count while you have the second count in your sight.

A better alternative would be some app that plays a click exactly once a minute. Put that on your earphones, start counting on one click, stop counting on the next click. The advantage is that it leaves your sight unhindered, allowing you to do it with a low amount of traffic.

For counting cadence, it's advisable to count double strokes. That cuts the counting speed in half, and if you remember your pedal position on start/end, you can deduce the half-stroke count exactly.

The magnet less sensors such as Garmin and Wahoo that are designed to the crank, attached to your shoe, might work. These use accelerometers to determine cadence. The difference when attached to your shoe are they stay the same orientation, when attached to the crank they rotate upside down every revolution, so it might not work as well.

A comment posted here indicates the Wahoo sensor might work for you (but which one? they probably have many variations)

I change between riding three different bikes, this allows me to only need one cadence Sensor. Out of the four different cadence sensors I own, this is the only one that worked consistently while attached to my shoe.

It could get expensive and you might end up with a pile of sensors that do not work - As suggested in the comments, using one with an ORing to attach to the crank would be fairly quick and easy to put on different bikes. For me I suspect that would get expensive as I would forgot to remove it when returning the bike.

Tabulate it.

If you know the number of teeth for each gear, and the wheel diameters, you can calculate your cadence at any given speed.

Make a spreadsheet or table with your speed in the columns in steps of e.g. 3 Km/hr or 5 Km/hr, and on each row list your gears, and then calculate your cadence for each gear at each listed speed.

An online calculator such as this one at http://bikecalculator.com/ may help you.

Print off your table, laminate it to protect against the weather, and attach it somewhere to your bike where you can see it without it distracting your attention.

Now if you are riding at, say, 22 km/hr, and your table shows a cadence of 60 rpm at 20 km/hr in that particular gear, and 65 rpm @ 25 km/hr, then you know your cadence is roughly about 62 rpm.

Please Note:

• Looking at tiny printed text in tables while riding is not a good idea. If you just have a small number of gears e.g. 5 or 7, you can make each row in the table sufficiently large to allow you to see the numbers without distracting you from the road.

• The OP asked about measuring cadence; this method is not a measurement per se, but rather a rough means of determining your approximate cadence.

a 3 axis walking pedometer placed in a sock works great. Needs no connection (powered by one internal AAA battery), but obviously must be manually removed from sock to read pace.

Many such pedometers are available via internet at prices around $20. I have used several different ones over 30 years of indoor and touring cycling, with all being extremely accurate at reporting total and average pedal revolutions.