New Mexico: A petty misdemeanor called "improper use of official symbols" occurs if anyone uses "the state or national flags for any purpose other than the purposes for which it was designed by law," if they "insult by word or act to the state or national flags," and if they use the national or state flags for advertising.

Penalty: none specified.

Source: 30-21-4

Analysis:
New Mexico's law "protecting" the American flag is especially broad in that it makes it a crime to use either the state or the national flag for any purpose other than what was intended "by law." This suggests that it's not just a crime to deface or burn a flag during a protest, but perhaps to use the flag as part of any protest.

It's a difficult question because it's not clear exactly what is meant when they say "for which it was designed by law."

New Mexico's law is also notable for the fact that it makes it a crime to not just to "insult" the flags by actions, but also by words. Using words is obviously speech and an "insult" is definitely a message, so it's clear that New Mexico intends to criminalize disfavored messages rather than to merely protect the physical integrity of flags.

More:

• Overview of State Laws on Flag Abuse, Burning, Desecration
• Index of State Laws on Flag Abuse, Burning, Desecration