What exactly is a "Cozy Mystery", and is there any stigma to them?
Traditionally, cozy mysteries have been written for older women, who would prefer there not be graphic violence, graphic language, or graphic sex.
The main character is usually an intelligent woman, with excellent powers of observation. She is not officially working with law enforcement. In fact, official law enforcment people tend to dismiss her actions as "play" or "unimportant." She often has a close friend or boyfriend who is a member of law enforcement, from whom she often gets information that is not available to the general public. The amateur sleuth is usually well liked, has a stable place in her society, and has flaws, but none that are "bad". (She can have a secret chocolate addiction, or can regularly skip her appointment with her personal trainer, but cannot be an alcoholic.)
Cozy Mysteries are intellectual challenges. The crime is a puzzle, and the main character (and reader) solve the mystery through their powers of observation and the ability to put the pieces together.
The crimes tend to be non-violent, and often take place "off stage", so they're not as distressing to read about.
There's usually no (or limited) off-color language. Sex may be implied, but is not usually described graphically.
By the end of the book the "bad guys" have been located and incapacitated, so society is back in balance.
Most of this sounds exactly like the Meg series by Andrews. Andrews' crimes tend to be a bit more graphic. Getting stabbed with a wrought-iron lawn ornament, or skewered with a branch of holly (with the resulting blood spatter) is more graphic that you'll often find in the mildest cozies.
However, her victims and perpetrators tend to be unpleasant gits. (The victim might get beaned with a statue, but you've probably been thinking about beaning the victim yourself...) The closest Andrews really came to having a sympathetic bad-guy was the murderer in Murder with Puffins. (No, no spoiler. Go read it yourself!)
One of the amusing side issues of Cozy Mysteries is they tend to occur in small cities or communities. And, since there's an average of one murder a year, the population gets thinned fairly quickly.
Imagine your terror at finding out that Jessica Fletcher was moving from Cabot Cove to your neighborhood! Would you stay up at night just wondering when this unassuming, friendly woman was going to befriend you? Would you wonder which of your friends would be the first, and then second, third, even fourth to die? I have loved watching Jessica solve all the murders in Cabot Cove, and then, when she ran out of neighbors, have to move to New York. (from the Cozy Mystery List.)Cozies have been evolving over time. Some have stronger language, more graphic crimes, and more visible sex.
My "live action" book group had been reading a series of heavy-duty titles, so we decided to change-up for a month, and read a Janet Evanovich Stephanie Plum book. One of the members is a charming, much-older woman, who has repeated assured us that any title is fine, as long as the characters don't "take the Lord's name in vain."
I'm afraid that Stephanie Plum pushed her envelope too far. When we met the next month, she was shaking with distress. The language and actions had been too much for her.
And yet, my own mother reads and enjoys the Plum series. (Even if my sister, my niece and I need to explain bits to her, which sometimes leaves her a bit breathless. She usually takes the information to her quilting group, where they all enjoy it enormously!) Would you classify the Plum series as "Cozy?"
There are a lot of sources for more information about Cozies.
The Cozy Mystery List has explanations, titles by theme, titles by author, a blog, and more.
Writing World has an excellent site that explains what you need to consider when writing a cozy.
The Madison Public Library has an excellent, extensive, and easily accessible list of cozy authors.
Is there a stigma to reading cozies? As most of Agatha Christie's mysteries could be classifed as "cozy," I think there's no stigma attached.