Rose names can be complicated and you have chosen an interesting example. For many plants, the cultivar name (indicating variants selected or derived by gardeners) would be expressed in Roman font*, in single quotes with initial capital letters. The cultivar name can follow the genus, a specific epithet, a hybrid or an infraspecific taxon name. For instance:

Primula auricula ‘Big Thrill’ or Rhododendron ‘Hoppy’

In the case of roses, many have been bred over decades to diversify their genetics and produce particular traits. The plants no longer have the strong characteristics of a particular species and are named simply with the genus, Rosa, and no specific epithet.

There is often also a cultivar name registered by the rose's breeder (called the registered name, breeders' name or cultivar name) as well as a name under which the rose is sold, known as a trade name (or trade designation or exhibition name). The American Rose Society serves as the International Cultivar Registration Authority for Roses and follows the rules set forth by the International Code of Nomenclature for Cultivated Plants ("ICNCP") in the registration of new rose varieties**.  A trade name is not a name regulated under the ICNCP,  but a device that is used for marketing a cultivar. A rose cultivar may be marketed under different trade names in different countries.

In the case of the rose Bring Me Sunshine, the registered breeder name (cultivar name) is ‘Ausernie’ and the trade name is Bring Me Sunshine. The trade name is not regarded as a direct equivalent of the cultivar name and should not expressed in the same way. The trade name and the registered breeder's cultivar name should both be indicated but should be styled in different ways; the trade name should be in a different font without quotes while the breeder's cultivar name is styled in the usual way. This particular rose is correctly expressed as

Rosa BRING ME SUNSHINE ‘Ausernie’

If the rose you are discussing has a trademarked name, that should be indicated with the addition of ® or ™ as appropriate following the trade name. There are also roses that are simply marketed under their registered breeder's name and do not have a separate trade name.

Rosa 'Evelyn May'

Sometimes you may see the use of the name Rosa × hybrida which appears to be an accurate botanical name, but it is actually just an indicator of a hybridized rose. It is an informal classification of roses that have been bred over decades to diversify their genetics and produce particular traits such as the sturdy, long-stemmed roses often used by florists. The plants no longer have the strong characteristics of a particular species. Here is an example of how you would accurately style the name of one version of red hybrid tea rose:

Rosa LEGENDS™ 'WEKsiamia'

There are no rules governing common names for roses. It is standard usage to write the word rose in lower case and to use Roman font but you will see the cultivar and trade names expressed in a variety of ways, before or after the word rose, depending upon the purpose.  As the first three letters of the registered name may offer information about the breeding of the rose, you will often see those letters written in upper case ('AUSernie', for instance, which indicates the breeder of the rose is David Austin Roses Ltd.).

For more information on expressing botanical names, see What is the proper way to write a botanical name? and Understanding Orchid Names.

Other resources:

Royal Horticultural Society Recommended Style for Printing Plant Names

International Code of Nomenclature for Cultivated Plants

Help Me Find Rose Search is a good resource if you know only the trade name for a rose and need to know if there is also a breeder's registered cultivar name.

Courtesy of NYBG Plant Information Service

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* In this explanation we use the expressions italicized to mean slanting font (or type style) and Roman to mean upright font.

**This registration system does not confer any legal protection over the plant or the name. Such protection has to be sought through statutory schemes such as national Plant Breeders' Rights ("PBR") outside of the US or Plant Patents in the US. A trademark protects the name of a plant but not the genetic material which can be protected by a PBR or plant patent