Are you restricting yourself to Sharjah only or is adjoining Dubai also in the picture?
Dubai's food scene is much larger and varied. Although Sharjah is a separate city and Emirate it primarily functions as a working class suburb of Dubai. A number of the popular Indian and Arabic Dubai "cheap eats" also have restaurants in Sharjah but the real action, so to speak, is in Dubai. I've lived in Dubai for five years now and have never met anyone who went to Sharjah just to eat at a certain restaurant.
A couple things to point out:
"Local" as in Emirati food is rare and when available is not remarkable. The default "local" food is Arabic/Lebanese. Very good selection of such places to pick from. Most of them taste the same but some are better than others. Al Mallah (locations both in Dubai and Sharjah) does excellent working class Lebanese fare and grilled meats. Al Reef (both in Dubai and Sharjah) does excellent Lebanese pastries, baked breats, manakishes (flatbreads with toppings). You will see corner "grills" all over the place with chicken roasting in outdoor rotisseries and with a large selection of fruit juices to select from. Cheap and fine for a quick lunch.
Indian and Pakistani food is stellar and many claim that it's better than in India or Pakistan.
Dubai is also a good place to try Persian food.
"Street" food just about doesn't exist other than food carts in the shopping malls.
For Dubai recommendations there's another thread in this forum.