Yes they are able to claim themselves as single on their own returns HOWEVER since you are claiming them as a dependent they need to indicate that they are claimed on someone else's return as a dependent.
You can claim them as a dependent as long as all the following are true:
Are they related to you? The child can be
your son, daughter, stepchild, eligible foster child, brother, sister, half
brother, half sister, stepbrother, stepsister, adopted child or an offspring of
any of them.
Do they meet the age requirement? They must be under the age of 19
or, if a full time-student, under the age of 24. There is no age limit if
your child is permanently and totally disabled.
Do they live with you? They
must live with you for more than 1/2 of the year.
Do you financially
support them? Your child may have a job and earn income, but that job cannot
provide for more than 1/2 of their support. You need to be providing for more
than 1/2 of their support even while they are working.
Are you the only
person claiming them? If you and your spouse are divorced, you need to be the
one claiming them for tax purposes.
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