The "degree Celsius" has been the only SI unit whose full unit name contains an uppercase letter since the SI base unit for temperature, the kelvin, became the proper name in 1967 replacing the term degrees Kelvin. The plural form is degrees Celsius.
The general rule of the International Bureau of Weights and Measures (BIPM) is that the numerical value always precedes the unit, and a space is always used to separate the unit from the number, e.g. "30.2 °C" (not "30.2°C" or "30.2° C"). The only exceptions to this rule are for the unit symbols for degree, minute, and second for plane angle (°, ′, and ″, respectively), for which no space is left between the numerical value and the unit symbol. Other languages, and various publishing houses, may follow different typographical rules.
Unicode provides the Celsius symbol at code point U+2103℃degree celsius. However, this is a compatibility character provided for roundtrip compatibility with legacy encodings. It easily allows correct rendering for vertically written East Asian scripts, such as Chinese. The Unicode standard explicitly discourages the use of this character: "In normal use, it is better to represent degrees Celsius "°C" with a sequence of U+00B0°degree sign + U+0043Clatin capital letter c, rather than U+2103℃degree celsius. For searching, treat these two sequences as identical."
Shown below is the degree Celsius character followed immediately by the two-component version:
When viewed on computers that properly support Unicode, the above line may be similar to the image in the line below (enlarged for clarity):