Most U.S. cities and towns use the International Building Code (IBC) with minor local amendments. The IBC exempts one-story detached shed under 120 square feet from a permit requirement; however, if a local amendment is different, say only exempting sheds over 60 square feet, the most restrictive rule applies. Regardless of the permit requirement, the construction must comply with code requirements regarding material, foundation and safety.
Your city may restrict the location of the shed. Larger sheds may require a setback from a side or rear property line. Some may be prohibited if the yard size does not provide enough useable outdoor area. Other rules may prohibit placement viewable from the street or other public right-of-way. Location regulations are generally included in a zoning code and often available online. Both building and planning departments often have a walk-up counter where you can speak with inspectors and planners about shed requirements.
Another common zoning restriction pertaining to sheds is height. Some codes may simply limit the shed to "one story." Others may more specifically limit the roofline to a certain height above grade. Different rules about size and use may apply if the height is above a given threshold.
Check if your city has adopted design guidelines under the authority of its zoning codes. These guidelines sometimes only apply to portions of the property that are visible from the public right-of-way but sometimes apply to the entire property. Design guidelines may specify exterior material and detail along with window and door placement.
In addition to local building and zoning codes, some homeowner's associations (HOA) have covenants, codes and restrictions that regulate the size, location and look of sheds. Check your copy of the HOA restrictions before building a shed.