What commands can be given in the Kindle's search box?

At least the following are supposed to work, although I haven't verified them:

;debugOn // verbose logging
;debugOff // non-verbose logging
~help
~usbNetwork  // starts a Dropbear SSH server
;ReadingTimeOff - switches off the reading time display
;ReadingTimeOn - switches on the reading time display
;ReadingTimeReset - resets the reading time computation
@author AuthorName keyword
@store keyword
@web keyword
@wiki keyword
@time

As @Tim pointed out in the comments, ~usbNetwork is non-standard, and some of the other "searches" on here may be hacks that require installation of additional software.

The following may be Touch-only; I can't verify this:

;dm - Dump messages to /documents
;dh - Dump cvm heap
;dt - Dump cvm stack
;shpm - set device to shipping mode
;urst - Reset user partition, deletes content of hidden System folder, Audible folder, Documents and tts folder. 
        Before using do a complete backup of your device
;debugPaint - log painting functions
;debugPref - pref level logging
;dP - alias of ;debugPref
;311 - change carrier settings
;411 - server information
;611 - wan information
;711 - wifi information
;fc-cache - updates fontconfig's cache, then restart the framework
;setTime - sets kindle time to unix clock
;st - alias of ;setTime (format: yyyy-mm-dd HH:MM – e.g.: ;st 2012-07-22 17:59)
~ds - Never show screen saver   (then you cannot lock the kindle till next reboot. 
                                 Rebooting the Kindle will restore the screen saver 
                                 lock and, hopefully, everything goes fine!)

The following are apparently Paperwhite-only, but I can't verify this:

";dm" : "/usr/bin/dm.sh",
";dmcc" : "/usr/local/bin/dmcc.sh",
";dh" : "/usr/bin/dh.sh",
";dt" : "/usr/bin/dt.sh",
";wifipopup" : "/usr/local/bin/wifipopup.sh",
";sandbox" : "/usr/local/bin/sandbox.sh",
";sbx" : "/usr/local/bin/sandbox.sh",
";shpm" : "/usr/sbin/shipping_mode",
";lzzl" : "/usr/sbin/shipping_mode",
";urst" : "/usr/sbin/userstore_reset",
";usbnetwork" : "/usr/local/bin/usbnetwork.sh",
";un" : "/usr/local/bin/usbnetwork.sh",
";di" : "/usr/local/bin/indexerdisable.sh",
"`stopIndexing" : "/usr/local/bin/indexerstop.sh",
"`startIndexing" : "/usr/local/bin/indexerenable.sh",
"`disableIndexing" : "/usr/local/bin/indexerdisable.sh",
"`indexStatus" : "/usr/local/bin/indexstatus.sh",
";ddc" : "/usr/local/bin/dynconfig.sh",
";resetConfig" : "/usr/local/bin/resetConfig.sh",
";rc" : "/usr/local/bin/resetConfig.sh",
";twoFingerChromeOn" : "/usr/local/bin/altChrome.sh 1",
";homeKeyChromeOn" : "/usr/local/bin/altChrome.sh 2",
";normalChrome" : "/usr/local/bin/altChrome 0",
";debugOn" : "/usr/bin/debugOn.sh",
";debugPaint" : "/usr/bin/debugPaint.sh",
";debugOff" : "/usr/bin/debugOff.sh",
";debugPerf" : "/usr/bin/debugPerf.sh",
";dP" : "/usr/bin/debugPerf.sh",
";311" : "/usr/bin/311.sh",
";411" : "/usr/bin/411.sh",
";611" : "/usr/bin/611.sh",
";711" : "/usr/bin/711.sh",
";setTime" : "/usr/bin/dateTime.sh",
";st" : "/usr/bin/dateTime.sh",
"~ds" : "/usr/bin/ds.sh",
";toggleLight" : "/usr/bin/light.sh",
";fc-cache" : "/usr/bin/fc-cache.sh",
";htmlViewer" : "/usr/local/bin/htmlViewer.sh",
";installHtml" : "/usr/local/bin/installHtmlViewer.sh",
";merchant" : "/usr/local/bin/merchant.sh",
";updateCamp" : "/usr/bin/updateCamp.sh"

Source

Here is an interesting tutorial on how to root your Kindle (not directly relevant to the question, in that it's not just a list of search commands, but relevant to Kindle hacking generally).

The commands ~ds and ;411 worked for my kindle paperwhite 3. The ;411 command even measures the temperature of the display! To re-enable screensaver for ~ds command, you must reboot the kindle paperwhite a few times. Mine took two times.

";dm" gets you the device logs, checked on kindle paperwhite 1, very useful

;dm worked for Kindle 10th Gen

I've seen this comment on Reddit that states that ~disableScreensaver and ~resumeScreensaver could work, but it didn't for me. It might be for older Kindles only. I have a Kindle Paperwhite 10th gen.