How long do the batteries on these scooters from Bird, Lyft, Lime last on a single charge? How far can I travel and for how long?

Definitely enough for an entire day. Max distance 20~30 miles, 2~3 hours of driving time, etc. on a single charge.

Assuming your scooter is freshly charged, the scooter should have ample energy to get you to your destination or even carry you on an all-day tour around town. 

Here's an example of a scooter from Bird.  (Remember there are many companies out there.)

Bird claims that Bird One can travel at speeds upwards of 19 miles per hour, carry a maximum weight of 220 pounds, and complete more rides on a battery in a single day than most competing scooters (up to 30 miles on a charge). Moreover, it expects that One will outlast modified consumer scooters by nearly 5 times and that its 473Wh battery will last twice as long. (source)

Of course, this is assuming your scooter is freshly charged. You can check within your scooter app.

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Uber Driver, CEO of TaxiFareFinder and Unleashed, LLC