First of all I suggest you reading the Technical Support Guide. Large portion of this guide is based on that thread.
I've categorized some common issues and listed possible solutions, but it does not mean they will always work.
I'm writing this based on my experience. The solutions worked for me so I'm sharing some tricks to make the game more playable.

I haven't really run into this problem myself, but from what I've seen, reinstalling PhysX solves the problem for most people.

EDIT: After installing the game on my new PC I got this start-up crash. Now I can prove that it works :)

Here are the steps:

Originally posted by Spoonparty:

-Go to where Deadly Premonition is installed, easily accessible by right-clicking the game in your Steam library > Properties > Local Files > Browse Local Files
-Open the folder named 'redist'
-Scroll until you find the file "PhyX_SystemSoftware.msi
-Double click
-Run the 'Repair' option
-Now run Deadly Premonition via Steam as usual

EDIT: Sometimes doing this will cause other games that use PhysX to freeze (eg. Borderlands 2). Once you've repaired PhysX using the installer in DP's redist folder, download and install the latest PhysX here[www.nvidia.com], and DP will still play okay. My best guess is DP uses a very particular DLL that only old versions of PhysX software contains it. Installing newer versions of PhysX won't remove it, so feel free to update your PhysX as usual.

This section only applies to C++ Runtime Error. If it's "DP.exe has stopped working" during game play, look into the next section.

It is difficult to completely eliminate C++ runtime errors, but here are a few things you can try and they should lower the chance of such error occurring:

• Disable the Steam overlay (Steam > Settings > In-Game > Uncheck "Enable the Steam Overlay while in-game")
  
  
  
• Use an application to edit DP.exe's File Header to allow it to handle >2GB RAM. (your machine must have 3GB of RAM of more)
  Here is an example of using said application: http://simswiki.info/wiki.php?title=Sims_3:Large_Address_Aware/CFF_Explorer
  Follow the steps but replace TS3W.exe with dp.exe in the Deadly Premonition folder.
  
  
  
• Use windowed mode instead of fullscreen. But if you really want fullscreen, you can use a mod called DPfix (download DPfix 0.9.5 here[blog.metaclassofnil.com]), but make sure to edit DPfix.ini and set borderlessFullscreen to 1. ALSO MAKE SURE THAT "Full Screen mode" IS UNCHECKED IN DPLauncher!!!
  
  
  
• Set "Run as administrator" in both DP.exe and DPLauncher.exe.
  
  
  
  
  
• While I don't think it is necessary, you can set compatibility mode to Windows 98. It may cause framerate issues.
  
  

Usually this happens when the game is loading the map at some particular time of day.
It can also happen right after a cutscene.
Sadly these crashes are caused by actual bugs in the game, and there is really no way to avoid them.



If you ran into these crashes, try the following:

• Try smoking for an hour or so before leaving the building.
  
• It's better not to save after the end of each chapter. Only save at phones.
  
• Backup your save often. It is in "savedata" folder. (Especially before chapter 9!)

THIS IS VERY IMPORTANT.
The game has a very frustrating bug that can crash the game (DP.exe has stopped working) right after the cutscene in the beginning of Chapter 9.

Please, backup your save before Chapter 9. Or if you are lazy, just don't save when it prompts after talking to Diane. If unfortunately you saved and don't have a backup, you might crash every time you load the save. Your only option would be starting a new game,

If you game crashed after the cutscene, load your backup save and smoke cigs for an hour.

If FPS drops during gameplay, simply use Alt+Tab to switch out and switch back in.
If the frame rate is still very bad, try restarting the game or/and rebooting the system.

1. Open Control Panel
   
2. Go to "Set up USB game controllers"
   
3. Select your Xbox 360 controller
   
4. Click "Advanced"
   
5. Change "Preferred device" from (none) to "Controller (Xbox 360 Wired Controller)"

Don't get confused between compatibility issues (since it's a port from Xbox 360) and actual bugs. They are different things. Compatibility-related crashes can be avoided if you follow the steps in this guide, however bug-related crashes will still occur.

At least for me, the game now rarely crashes.
If I missed anything please leave a comment and let me know.

I think this game is a masterpiece, and I hope everyone can have a chance to enjoy it.

Technical Support Guide by Spoonparty
Durante (Author if the DPfix)
Jackpot
Chronicler
NTCore
James D