Handbag falls in Class 18, bags & leather goods, and depending on what exactly you sell it can also fall in Class 26 or Class 9 or Class 11. Everything below is read from the USPTO's own list of accepted descriptions, not from an opinion about where it belongs.
Searching the USPTO Trademark ID Manual for handbag returns 48 accepted descriptions, and most of them sit in Class 18. The manual is the list of identifications the office has already approved, so wording taken from it is pre-approved and settles the class at the same time.
The class follows from the identification you write, so "what class am I in" is really "what exactly do you sell". How the class system works covers the rest.
Luggage, handbags, wallets, or other leather goods (and umbrellas).
37 accepted descriptions in the manual mention handbag. For example:
Sewing notions, lace, ribbons, buttons, or hair accessories.
4 accepted descriptions in the manual mention handbag. For example:
Software, apps, downloadable content, computers, or electronic devices.
4 accepted descriptions in the manual mention handbag. For example:
Lighting, heating, cooling, cooking, or plumbing equipment.
1 accepted description in the manual mention handbag. For example:
Handbag appears in 4 classes because what you make, what it is made of and the service of selling it are separate classes. Pick the descriptions above that match what you actually do.
Making handbag is Class 18; a shop or online store that sells it is retail store services (Class 35), and a business that does both files in both.
Bags, luggage and leather goods: handbags are here. Leather is not the test, though. Leather jackets are clothing, so they are Class 25.
The class decides what a registration covers; it does not decide whether the name is available. Relatedness is judged on the identifications, so a conflict can sit in a class you never looked at: the same name in a different industry.
Check whether your name is clear for handbag Searches the live USPTO register, including sound-alike and meaning-alike names. First search $2, then $5.Handbag sits in Class 18, bags & leather goods. That is where the USPTO's own ID Manual puts it: 37 accepted descriptions mentioning handbag fall in Class 18. Depending on exactly what you sell it can also fall in Class 26 or Class 9 or Class 11.
You file in every class you actually use the mark in, and the USPTO charges its fee per class. Selling a product and also running an online store for it is two classes, not one. Filing in classes you do not use is worse than filing in too few: an application has to be supported by real use or a genuine intent to use.
No. The class decides what a registration covers. Availability turns on whether someone already holds a similar mark for related goods, and relatedness is judged on what the identifications say rather than on the class number, so marks in different classes can still conflict.
Not what you sell? Search the class finder for your own wording, or browse all 45 classes.