If you had to select the most trapped character in the text, who would it be? Why?

The Handmaid's Tale is a story about various classes of men and women, all of whom are trapped, to some degree, by the class they are born into.

The wives in the novel are trapped by their infertility. While they live in the hierarchy of society, they belong to their husbands and are controlled by them. Serena Joy is not allowed to enter the Commander's study, a private place where he permits visits with Offred, but where his wife is forbidden. She is not allowed to read or write, forced instead to remain uneducated and unaware of the realities of the world around her.
It's simplistic to choose Offred as the most trapped character in the novel, although she is undoubtedly a victim of circumstance much like the other characters. It is implied that Offred could be employed at Jezebels if she were unwilling to complete the jobs of a handmaid, an opportunity not offered to Serena Joy. In fact, women employed at Jezebels are allowed to use drugs and alcohol and to refuse relations, if they wish. These freedoms would be available to Offred, if she chose that job, and would never be made available to Serena Joy.
A clear argument could be made for the lack of freedom of any of the women in The Handmaid's Tale. However, it's Serena Joy's perceived freedom that makes the fact that she is trapped that much more poignant.

This is a really tough question to answer, because all of the characters are trapped to some extent. Even men like the Commander are limited and must abide by certain rules and conventions in Gilead. If I had to choose one character as the "most trapped," I think I'd select Serena Joy.
Serena Joy holds the illusion of power but really has none. The wives are in an impossible situation: they are apparently unable to bear children (though the text hints that it's probably male infertility in many cases) and must welcome strange women into their houses to be impregnated by the husband, the Commander. The wife must take part in "the Ceremony" in which the husband has sex with the handmaid while the wife sits with her legs around the handmaid and holds her hands, as though they are one woman. The wife is the one who benefits when the handmaid gets pregnant and has the child, though she cannot and does not go through the process of pregnancy and giving birth. If the handmaid is unable to conceive, the wife really has no power or control to do anything about that. We see Serena Joy try to circumvent the system by telling Offred to sleep with Nick, but we can also tell that she judges Offred for doing so. Serena Joy seems the most trapped to me because there really is no scenario in which she can be satisfied in this society.
Yes, Offred is certainly trapped, but since we get the story from her perspective, we see her as more human and more rebellious. We know what she is thinking and how she wants to try to transgress or escape the societal rules for handmaids. Serena Joy, however, seems bitter and hopeless. She lives in a society that sanctions and even requires her husband's infidelity while she stands by and has to accept it. For these reasons, I see Serena Joy as the most trapped, even though she is technically near the top of the society's hierarchy.