She's hoping
i think
was wondering
i expect
they're trying

these all come under the category of progressive verbs? If not can some one please exaplin which are wrong? I have found them in an extract and want to know if i have chosen the right ones.

thank you for any help!
Yes, they can all be used in progressive aspect.
  1. She's hoping = She is hoping >> present progressive (aka present continuous)
  2. I think >> simple present (NOT progressive)
  3. I/he/she was wondering >> past progressive (aka past continuous)
  4. I expect >> simple present (NOT progressive)
  5. they're trying = they are trying >> present progressive (aka present continuous)
#2 and #4 are not in the progressive form.

A progressive form of a verb consists of the verb "be" (in your examples: is, was, are) followed by the -ing form of the verb you're using (in your examples: hoping, wondering, trying)

You can read a very simple explanation of the present progressive here and of the past progressive here .
Other tenses may have a progressive form, as well. For instance, you may have: present perfect progressive (I have been waiting), past perfect progressive (I had been waiting), future progressive (I will be waiting), future perfect progressive (I will have been waiting).

Does this help?
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Thank you very much, exactly what i needed Emotion: smile
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One other question, if i had to identify the progressive verb groups in "I think she's hoping" it would simply be the "she's hoping" segment, yes?
Yes. There is an optionally omitted 'that': I think (that) she's hoping.