If you search using Spotlight (from the Home screen), you can use
from: qualifiers, as well as flag and unread status.
From Apple's iOS: Understanding Spotlight Search page:
Flagged june 2013
Searches for all flagged messages from june 2013, as well as for messages containing these words
Unread last week from:@icloud.com
Searches all unread messages from last week where the sender's email ends with @icloud.com, as well as messages containing these words
Vip read yesterday
Searches all read messages from contacts marked as VIPs, as well as messages containing these words
Searches all messages sent on Monday to a contact named "Appleseed", or an email address starting with appleseed, as well as messages containing these words
Note that these examples mention "messages containing these words". A note above the examples states that to search message bodies, you must use Mail's search function, not Spotlight, but these examples are for Spotlight. After testing, it appears these work in Mail's search field as well as Spotlight.
In addition to the above, you can use
has:attachment to limit to messages containing attachments, and Mail appears to search within attachments.