I have found that granting permissions of Copy, Export to User, Create and Save search to User on a Role configuration gives the role/user with Read Only access the ability to create and New Search. However, it also gives the ability to Copy Assets and additional functionality that a Read Only role/user should not have access to.
Design change with the 14.1.02 Cumulative patch
A role/user with Read Only access should neither be able to create a search nor be able to create assets. Originally in 14.1 these permissions were different however, it was decided that changes in the functionality needed to be handled.
A Read Only role/user should not have the ability to create assets.