Where are Android logcat files stored?

They are stored as circular memory buffers on the device. If you run “adb logcat > myfile” on your host system, you can retrieve the content into a file.

To get post-mortem information from a device it is helpful to see the timestamp as well. To let you know when the event occurred, try:
adb logcat -d -v time > logfile.txt

If you just want to see the current contents of the log, you can do
adb logcat -d > myfile

It will exit after dumping the log.

(suravi’s suggestion would be good if you wanted to watch the log while testing.)