Euonymus scale is a good guess, but this looks like Cottony Camellia Scale, which can get on Euonymus.
Cottony Camellia Scale is a soft scale, and soft scales suck plant juices and then excrete a sticky substance called 'honeydew', which the black film called Sooty Mold grows on.
(Euonymus scale look different and are an armored scale, which don't make honeydew.)

You can see photos of each and learn about control options on this page: 
https://extension.umd.edu/resource/introduction-scale-insects

Heavy or prolonged infestations can cause leaf yellowing and a decrease of the plant’s vigor, but dieback is not common.
Euonymus can take a pretty hard pruning in early spring and respond with fresh, new green leaves.

If you are not in love with them, there are other plants that are better for the environment and less likely to have this issue.


Christine

Thanks for the quick response. I am not at all attached to the euonymus. I inherited them when I bought the house 17 months ago. What would recommend I replace them with? The area is quite shaded, mostly by a large tree. 

John

Fortunately you have a range of options. It's difficult to make specific recommendations without more information (space available, desired mature size, whether deer browsing is an issue, desirable features like flowering or evergreen, etc.), but given the shady exposure, here is a starter list of ideas:

• Leucothoe (Leucothoe fontanesiana, Leucothoe axillaris) - native near our region, though not in Maryland specifically; evergreen; deer-resistant
• American Holly (Ilex opaca), specifically variety 'Maryland Dwarf' because no other variety is small enough - easy to trim but shouldn't need it often due to slow growth rate and compact size; evergreen; deer-resistant; although it could produce a few decorative berries, this variety doesn't fruit heavily
• other Hollies (Ilex, multiple species and hybrids) - although they only other shrubby evergreen holly is more sun-loving (Inkberry), several options exist among the Chinese, Japanese, Blue, and other hybrid varieties; if dry soil is a concern, Chinese hollies have the best tolerance within this group; all are evergreen; deer only tend to avoid the very prickly varieties; not all varieties will produce berries
• Illicium (Illicium, several species and hybrids) - look for those rated cold-hardy to zone 6 ideally, but those hardy to zone 7 should suffice if this site is sheltered from winter winds; evergreen; deer-resistant; some species are native, though not in MD
• False Holly (Osmanthus heterophyllus) - prickly holly look-alike with fragrant autumn flowers; evergreen; deer-resistant
• Yew (Taxus) or Japanese Plum Yew (Cephalotaxus) - one of the only needled evergreens to tolerate shade; the former deer will eat, but the latter they dislike; several cultivars exist for each with different growth habits and mature sizes, though Yew is pretty easy to trim if needed
• Summersweet (Clethra alnifolia) - blooms best with more sun, but should tolerate shade if not too dark; fragrant summer blooms; deciduous; somewhat deer-resistant; dwarf varieties exist if the wild type is too tall
  
• Spicebush (Lindera benzoin) - fragrant foliage when bruised; deciduous; native; better for large spaces as it can't be kept too small via pruning
• Fothergilla (Fothergilla major, Fothergilla gardenii, and their hybrids) - native to the southeast but hardy here; deciduous; blooms best with partial sun but tolerant of shade; somewhat deer-resistant; not easily trimmed but compact varieties exist
• Carolina Allspice (Calycanthus floridus) - native to the southeast but hardy here; as the above, blooms best with a bit of sun but tolerates shade; very fragrant flowers; somewhat deer-resistant; not easily trimmed for size, so best where it will fit with minimal pruning

We suggest that, if space allows, you mix species so the planting isn't as vulnerable to a single ailment affecting the entire garden bed equally severely. Diverse plantings (especially using flowering shrubs or perennials) also tend to provide habitat for a wider array of beneficial insects, which help to suppress pests before they reach outbreak level.

Miri