iOS 7 iMessages stuck at "Sending Message"?

Are you seeing your iOS 7 iMessages stuck at "Sending Message"?  This means the messages have NOT been delivered!  Apple has promised a new (7.03) patch (as early as this week) to fix the issue; meanwhile, here's what you can do right now:

a) Reboot your iOS 7 device by simultaneously holding down the "On/Off" and "Home" buttons at least 10 seconds until your phone shuts off AND the Apple logo appears (signifying a restart).  This will force all the "Sending Messages" to at least show a "Not Delivered" notice; at this point you can choose to re-send (or not resend) the message.

b) Another change from iOS6 to iOS7 changed a default setting which would re-route iMessages via standard SMS text message (where sent message 'balloons' appear in a different color than the more secure iMessages).  With iOS 7 this is turned OFF -- good for folks who have a limited number of text messages/month.  If you have unlimited texting, you should turn this back on:  SETTINGS - MESSAGES - Send as SMS=ON.   Note:  Just turning SMS Texting back on as a 'fallback' to iMessage doesn't fix the basic "Sending Message" problem.

Having Trouble Sending iOS 7 Messages? Apple has a Fix on the Way
http://www.macobserver.com/tmo/article/having-trouble-sending-ios-7-messages-apple-has-a-fix-on-the-way

Finally, certain articles are advocating that certain users also need to "Reset network settings".
https://www.apple.com/support/iphone/assistant/wifi/#section_5

Do NOT do this unless you're fully aware of its SIDE EFFECTS !  According to Apple, "This will clear your current cellular and Wi-Fi network settings, including saved networks, Wi-Fi passwords, and VPN settings."

This means you will ALSO need to re-enter ALL Wi-Fi passwords you use (home, office, other frequent locations) before you again can use AirPrint printers, and until re-entered, you'll also use up more of your expensive cellular data plan instead of free WiFi.  "Reset network settings" will also erase any office VPN settings, and MAY erase any Bluetooth pairings you've setup -- meaning they too have to be re-setup.

Because of these side effects, I haven't yet resorted to "Reset network settings" or any other tips.  My recommendation is to again use the above iDevice restart procedure until Apple releases an iOS 7 update to fix the problem.