Thanks for letting us know and for the valuable warning about the ground wire for other MINI owners!

Got the problem taken care of , it was the DME freaked out. Mini of Orlando checked it and told me it was the CAS module although the scanner was reading the CAS and not the DME. I wasted a lot of time and money because I sent the CAS to one company to re-manufactured and it was returned as not defective. I then sent the DME to another company who returned it as not defective. I took the car to a BMW shop who installed a DME the had and got a reading so they verified it was the DME that got damaged. The company wanted to charge me over $2000+ to install a new one but I sent the CAS, Key and DME to Precision ECU and purchased a plug and Play Reman DME and it all Good.

WARNING: Never leave the coil ground wire on the front of your valve cover off and start your car. This is what caused the Hell I went through with my car. You would think the Manufacturer would have some inline protection since the placed it in such a stupid senseless location knowing what can happen if it is forgotten off.

This is pretty late, but the ground being off or loose was ground out and damage the DME. Looks like you need a DME replacement.

Schematic does not break out pin numbers for the relays. Best I can tell, the ground from the relay is a grey wire connected to pin 28 of the DME and is used to trigger the relay.

Pin 85 of the DME Relay located in the engine fuse box. Pin 85 is the Ground terminal for the relay.

I looked in the Bentley schematics but could not find a pin 85 on the DME...

I have narrowed down the problem to a bad ground, now I just have to find where it is broken. My DME relay kept clicking off and on  and when I tested terminal 85 it was not receiving a constant ground. I connected a ground wire from that terminal to the engine and the relay stopped clicking off and of but the engine is still cranking but not starting. Guess I have to get a wiring diagram so I can trace the route of that terminal 85 wire to find my problem. I will update this post when I find it because it seem a lot of Mini owners has been fighting the same issue with no solution. The mechanics at the dealers can only switch parts out hoping it solve the problem so please don't waste your money there like I did.

all those codes are related to the engine start operation

Unfortunately the diagnostic codes can't tell you 100% what the problem is, they only know their tiny piece of the larger puzzle - as an example:
the AOB3 code denotes starter malfunction - AOB4 is also related to starting but specifies the CAS module, and triggers when the diagnostic system notices that the car isn't starting. really vague.

3 month after and my 2010 Mini Cooper S is still down. I took it to the dealer who told me they got trouble code A0B3, A0b4, A0b6 and the problem was I needed a new CAS module for $550.00 and $775.00 to replace it because it takes about 3 hours to remove it and 3 hours to install. I took my car Removed the CAS module in 20 minutes and sent it to a Company to re-manufacture it. I was told the CAS was not defective it may be the DME so I sent that to be re-manufactures and got the same result it was fine No issues. If you have this type of problem Forget the Mini Dealer they are guessing just like you.
I bought 3 different Diagnostic Code readers and surf the web and no one has a proper explanation  of the or solution for 4 trouble codes I found, A0B3,A0B4,A0B5,A0B6. The last Diagnostic tool I purchased was a AP200 and the manufacturer could not explain the true fix for the codes their reader is giving.
Why manufacture a tool you do not know 100%.
I have given up on this junk.
I am at the point of taking my piece of junk and selling it to a junk yard. I will never buy another Mini Cooper.

Tried it all still no sparks Guess I really have to replace the ECU. I will sell this crap as soon as it is fixed, Done with Mini.

Keep us updated...

I will replace both relays with new ones and I have just purchased a new crank position sensor which I will also replace because I notice when I crank the engine, besides no sparks, the tachometer needle do not move period. I will check this tomorrow when I get the crank sensor in.

I know you mentioned that you checked all the relays and fuses.  But did you confirm that the 2 relays (side by side) in the box in the engine compartment are working properly.  You cannot swap them as they are both needed to keep the car running. I really doubt the open ground damaged coil packs or the ignition module.  I know I have accidentally left it off on both of my R56's. 

The ignition pulse to the coil comes directly from the main computer (ECU DME) in the engine bay....

The ignition coils is new,I even tried my old coils to check if the new ones were burnt out but it still did not start and no sparks.
I was thinking I may have to purchase a ECU DME Engine CAS Ignition Key Set , it will be cheaper than taking my car to the dealer to trouble shoot. I took it to the dealer when I had the misfire on Cyl #3 and they took my money and told me nothing I did not know.

Any idea where the ignition module is located? I did some search but came up with nothing.

You may have lost the ignition module, and there be tests of the module that would confirm that.

I rebuild my 2010 Mini Cooper s engine after the rear of #3 piston was broken off.
I test drove the car and heard a slight tapping noise so decided to remove the valve cover to check that all is good and forgot to reconnect the coil wire ground at the front of the valve cover. When I started the car again it start and close off then wiill not start again. The car cranks fine but will not start and there is no sparks at the plugs. I can not find a solution for this problem. I know leaving the coil wire groind off caused the problem but I tested all fuses, relays,sensors nothing. Can anyone help?
Thank You in advance.