In my application, few web forms have list box controls. When I hover mouse over the menu control, which is on master page, the list box control flickers. It shows kind of multiple borders for a very short period of time.

Sounds like you may have some kind of JavaScript hover function attached to your menu control.  It may be doing some weird stuff, especially if z-order CSS settings are involved.  DropDownList and ListBox controls have been known to clash with other controls in the past as far as UI is concerned.