These techniques have worked for us but there is no guarantee provided. Use at your own risk.
Always make a backup of important files before attempting any troubleshooting.
 

If FCPX won't launch or hangs or crashes on launch, the first thing to try is to remove all of your FCPX Events and Projects and unmount all external drives and try to launch Final Cut Pro X. I've had problems launching FCPX with all Events and Projects removed but with multiple external drives mounted. See details here.

1. Update FCPX to version 10.0.8 via the App store.
This update includes bug fixes and improves FCPX performance and stability.

2. If you have issues updating FCPX from the App store, try these steps from Apple:

(I had an issue with the App store not recognizing the account which I was logged in to as the account that I bought FCPX from after deleting the app. I luckily made a copy of the FCPX app on removable drive and once I moved that copy back to the Application folder, in the App Store I was able to download the update. My advice, make a copy of your Final Cut Pro X app on a removable drive or flash drive before you delete FCPX.)

Important: Before attempting this process, ensure your system has a graphics card that meets system requirements for Final Cut Pro X, Motion 5 and Compressor 4. Graphics cards that do not meet system requirements are listed here; even if your current graphics card is working with Final Cut Pro X, please check the list first before deleting your application.

If the update is not listed under the Updates Tab:

1. Drag the FCPX application to the Trash from the /Applications folder.
2. Empty Trash by clicking Finder > Empty Trash. 
....Note: This will permanently delete anything you have stored in the Trash.
3. Open the Mac App Store.
4. Click the Purchases tab.
5. Click Install for Final Cut Pro.

Reinstalling the application does not affect your Events or Projects since they are stored in a separate location.

Apple article link:
http://support.apple.com/kb/TS4057

Apple - Final Cut Pro X: How to remove and reinstall

Apple - How to remove the Final Cut Pro X Trial

Final Cut Pro X trial: FAQ

3. Make sure your machine and graphics card meets the Minimum System Requirements. The graphics card requirements for Final Cut Pro X are:
OpenCL-capable graphics card or Intel HD Graphics 3000 or later and 256MB VRAM

http://www.apple.com/finalcutpro/specs/

http://support.apple.com/kb/ht4664

Some graphics cards are no longer supported by the 10.0.5 update. If your card is no longer supported, FCPX might not install or operate properly.

4. Back up your Project Library, and Events before you start troubleshooting. Backing up should be part of your everyday workflow.

Apple's FCPX help section on backing up:

http://help.apple.com/finalcutpro/mac/10.0.3/#verb6acb2fb

FCPX now creates an automatic Backup file that is updated every 15 minutes with any changes you've made to your Project or Event. This file will be used when launching FCPX if the original Event or Project file is corrupted.

5. Try Hiding events and projects and then launch Final Cut Pro X  http://support.apple.com/kb/HT4740

6. Trashing FCPX preferences can fix some FCPX problems.

Make a backup copy on a flash drive and then delete the FCPX preference files located here:
User > Library > Preferences > com.apple.FinalCut.plist and
User > Library > Preferences >com.apple.FinalCut.LSSharedFileList.plist.

And for Lion also delete:
User > Library > Preferences > com.apple.FinalCut.plist.lockfile

Quit FCPX, backup and trash these preference files and relaunch FCPX. Or use the very handy and free Preference Manager 3.0 to backup and trash your FCPX preferences. Once you get FCPX running smoothly, make a backup of your preferences for future restore with Preference Manager.

7. Some symptoms can be fixed by simply quitting and restarting FCPX or by shutting down your Mac and rebooting.

8. Slowness (spinning beach ball) or hangs in FCPX, launch Activity Monitor to see what is taking up your system resources. It is located in your user > Applications > Utilities folder.

Pay special attention to Free memory and application % CPU usage. Low Free memory or an application with high % CPU numbers could be causing the spinning beach ball. Shut down most other applications while using FCPX to help if you have this problem. Two helpful articles about using the Activity Monitor are located here and here.

9. Close windows that you are not using in Final Cut Pro. If you are not using the Inspector, Scopes, Multicam Angles or the Event Library, close those windows.

10. If FCPX won't launch or hangs on launch, first try trashing FCPX's Preferences, #4 above. If that doesn't solve your problem, try removing all of your Events to eliminate the possibility of a corrupt media file. FCPX will create a new empty Event on launch. Your Events are located at the top level of your media drive in the Final Cut Events folder. Do this by moving your Events to another folder at the top level of your media drive. If FCPX then launches OK, one of your Events could be causing the problem.

Try playing your Event media files in Quicktime. If you find one that won't open, that is your suspect. Remove it and try launching FCPX again.

Add your Events back one at a time and relaunch FCPX to try to determine the culprit. If you find a problem Event, try removing 50% of the media from it and then try to launch FCPX.

Continue to use the process of elimination by removing/replacing media to try to locate any specific problem causing media files.

If working with Events doesn't solve your problem repeat the above process of elimination with your Projects. FCPX Projects are located at the top level of your media drive in the Final Cut Projects folder.

You can also buy Digital Rebellion's Pro Maintenance Tools, a very comprehensive suite of maintenance tools for FCPX, that includes an FCPX Crash Analyzer, Project Repair and Corrupt Clip Finder.

There is an easy way to manage your FCPX Projects and Events.

I use Philip Hodgetts very affordable and effective $5 Event Manager X to manage my FCPX Events and Projects before I open FCPX. You can select and deselect Projects and Events to open and work on in FCPX. And you can index all of them.

Event Manager X will even launch FCPX, or quit and relaunch FCPX, to activate your selection.

11. Try to only open the Events and Projects you will be using during your current session.
Opening too many Events and Projects, especially ones that you won't be working on, can bog FCPX down. Once again, Event Manager X works well for managing FCPX Projects and Events. Or you can manage your Projects and Events by manually moving unused ones out of their folders before launching FCPX.

12. Trouble with Sharing/Exporting? Use a process of elimination to zero in on the problem. Try sharing one half of your timeline at a time. If one half will export but the other will not, you probably have a corrupt media file or render file in the failed half. Try removing media to narrow down the problematic clip(s). Also try sharing another timeline in another Project to see if your sharing problem is just with a certain timeline.

13. A corrupt Project or Event render file could be a problem. Delete Event Render files by selecting the Event and then going to the File menu and choosing File > Delete Event Render Files. Choose All Render Files if you suspect an Event render file is causing your problem. Of course you will have to re-render everything if you choose this option.

A Project can also have its own render files so repeat the process for Projects. Select the Project and then go to File > Delete Project Render Files.

14. 3rd party Plugins, Titles, Generators, Transitions and Effects can cause FCPX to hang or crash. Some of them are located in your user > Movies > Motion Templates folder.

Start by removing all 3rd party folders there and see if the FCPX problem still exists. If it does not, try adding your 3rd party effects back one at a time to find the culprit(s).

15. If FCPX stalls or seems to freeze, cancel all Background Tasks. What does Background mean? Background Tasks aren't always background in FCPX.

I have found that Background Tasks in FCPX can overwhelm foreground tasks. They don't always operate in the background, as you might expect a background task to do, but will actually prevent, freeze or delay foreground tasks. They are supposed to pause when you are using FCPX but I've found this only happens with some tasks. For example, when background tasks are running on my 2011 - 3.1 GHz Intel Core i5 iMac with 12 GBs of ram, I can't rename an Event or Project in the foreground. I have to wait until background tasks are finished or else cancel them.

If you experience freezing or stalling in FCPX, and Background Tasks are running, my advice is to cancel Background Tasks until you want to engage them or when you take a coffee break.

16. Try to replicate the problem when logged in as a different User. If the problem goes away, the problem is in your original user folder. Also try testing FCPX with media that you know is OK from another project.

17. Clear potentially corrupted caches and fix some system problems by performing a Safe Boot. Hold down the Shift key while starting your Mac until you see a gray progress bar under the Apple logo. Safe Mode is the state Mac OS X is in after a Safe Boot.

Starting up into Safe Mode:
a) Forces a directory check of the startup volume.
b) Loads only required kernel extensions
c) Disables all fonts other than those in /System/Library/Fonts
d) Moves font caches to the Trash.
e) Disables all startup items and login items.
Reboot after Safe Mode finishes starting up and try FCPX.

You can also try Repairing Disk Permissions

18. If you have Perian installed, try uninstalling it from System Preferences. I haven't experienced any Perian specific problems with FCPX but Perian has caused Mac video problems in the past. You can always reinstall it if it isn't causing a problem.

19. If all of the above techniques fail to help with your FCPX problem, try a fresh install of FCPX.

Apple - Final Cut Pro X: How to remove and reinstall

After FCPX finishes downloading and installing, make a back up copy and store it on a removable hard drive or memory stick, for future troubleshooting. Apple recommends not having more than one copy of the FCPX app on any mounted partition.

Apple's Final Cut Pro X: Troubleshooting basics article:
http://support.apple.com/kb/TS3893

20. Visit the Final Cut Pro X Apple Support page or Support Forums
for additional troubleshooting advice.

Apple's Final Cut Pro X: Troubleshooting basics article
http://support.apple.com/kb/TS3893