Unable to profile app on device with iOS 9.0.1 using Xcode 7, 7.0.1 or 7.1 beta

TL;DR - Perform a complete reboot of your device; restart Xcode & instruments; select "Open Xcode" if prompted to enable the device for development.

Update 3/31/2016: I haven't encountered any issues with the latest version(s) of Xcode (7.2.x, 7.3), so it seems that the stability here has been improved.

I believe I may have finally gotten this to work properly. Detailed steps:

1. Unplug the device from your Mac & power down the device completely (hold the power button for several seconds; slide to power off).
2. Close Xcode and Instruments.
3. Restart the device & once it has booted completely re-connect it to your Mac.
4. Re-launch Xcode. Here, my device showed as disabled and Xcode indicated that the device was not available for use.
5. Open your project; clean (Shift+Command+K), Build (Command+B), Profile (Command+I).
6. After Instruments launched I noticed that the device was enabled. Upon selecting it, a message was displayed with the title "Enable this device for development?" and message "This will open Xcode and enable this device for development." (Note that this only happened to me the first time I went through this process even though I had already been using the device for development - whereas some users have also reported that they are not presented with this dialogue.)

7. Click "Open Xcode". Here Xcode did not prompt me for anything nor was anything displayed - no additional messages indicating anything had been done or that the device was or was not available for development. Opening the Devices window, the device appeared to be available. (I have not been presented with this option for subsequent occurrences.)
8. Now I was able to select the device in Instruments and profile it.

As a side note, I was also again able to delete installed apps from the Devices window (I realized that this was not possible to do previously).

I'm unsure how my device ended up in this state however I will be on the lookout to see if this continues to occur.

Please note that this was done using Xcode 7.0.1.

Update: My device seems to lapse back into not being able to be used for profiling some time after performing these steps - I've had to reboot my device again in order for it to be available for profiling. Not sure what is triggering this behavior but I will file a Radar for this.

Close instruments -> reset your device by long pressing home and power button -> restart instruments. Works for Xcode 7.3.

According to Apple staff on the developer forums the behaviour of this issue is greatly improved in the Xcode 7.3 betas, so one easy solution is to try upgrading Xcode. I've upgraded and it seems to be behaving so far.