Hey jasun,
Sounds like you've had a rough time!
I work for halifax, most highstreet banks give basic accounts to anyone just as long as your not looking for an overdraft facility if you've struggled financially in the past.
To open an account they will create a credit search but it's just standard as it a way of identifying you and checking for fraud.
Also a bit of advice for you as you have had someone using you details, you'll need to protect your information so they can't apply for anything with your details. Contact experian or equifax to put a "waring flag" on your credit file with a password so if you or anyone does try to apply with your details the lender will contact you to confirm it first.
Also regarding your financial history with you ex husband you can also create something on your credit file called "A Notice Of Correction" with this you can explain why you have been in a sticky situation and that it is not the normal way you would conduct your affairs.
If your ex husband had bad debt it can be affecting your hence not being able to get an account, you can dissociate yourself with him by speaking with the above companys.
08702416212 - experian
Hope this helps!