I am delightfull to announce you that I finally manage to jailbreak my iPhone 5S 32Gb running iOS 8.3. Here it is how I did it:

iOS 8.3 jailbreak step by step

Step 1: Download the jailbreaking tool released by TaiG from their official website (download iOS 8.3 jailbreak)
Step 2: Download iTunes 12.1.2 for older videocards.
Step 3: Remove any previous installed version of iTunes.
Step 4: Install iTunes 12.1.2 for older videocards, the one you just downloaded at step 2.
Step 5: Disable PassCode on your iPhone device. Disable Fingerprint if you have it Enabled. Also, disable Wi-Fi and any mobile connection 3G/4G (you will enable them later)
Step 6: Start iTunes and connect your iPhone to you computer using the USB Cable. Make a full backup of your iPhone before starting the jailbreak process (just to be sure).
Step 7: Unzip the jailbreak file (some antivirus programs might find it as a security risk. Please ignore the warning. I have tested the file and looks pretty safe, I have manage to jailbreak my iPhone and works without a problem so everyhting is just fine)
Step 8: Follow the instructions provided by TaiG’s jailbreak software and wait for your iPhone to reboot.
Step 9: If you’re jailbreak get’s stuck at 20% just repeat the process (it happened to me at first and I was on the point to quit but I have tried again and the jailbreaking process completed successfully).
Step 10: Once your iPhone rebooted, open the fresh installed Cydia and search TaiG. You will find the jailbreak tool (should be version 2.1.3 for now) and reinstall it. Basically this step is not necessary but in my case (running TaiG 2.1.2) I got issues with Mobile Substrate when I have received call or when I was trying to make a new phone call. Mobile Substraged was crashing and the iPhone entered Safe Mode and I hade to respring.

At this point you should have a jailbroken iPhone running iOS 8.3. I have tested a lot of tweaks from Cydia, restarted the phone, used 3G, used wireless. Everything seems to be in order, no more Mobile Substrate crashes.

iOS 8.3 jailbreak issues

Issue no. 1: After jailbreaking my iPhone, if I go to Settings -> General -> Software Update it shows Checking for Update but nothing happens. It doesn’t say anymore that I am running the latest version of iOS 8.3. I do not know if this is an issue with the jailbreak itsefl or the problem appeared because of some Cydia tweaks and apps that I have installed for testing and then uninstalled.

Issue no. 2: A lot of porn ads are displayed on Cydia especially when getting applications from the BigBoss repository. Also a lot of porn ads are displayed on Safara while browsing (I am not sure if this is a jailbreak issues because I had this problem even before jailbreaking, right after I have updated my iPhone from iOS 8.2 to iOS 8.3