A) rude to fail to invite the spouse or established partner of someone you’re inviting to a set-guest-list event like a wedding; and
B) rude to bring along to a set-guest-list event like a wedding anybody who wasn’t explicitly invited to attend.
The most serious problems occur when guests whose hosts have committed Rudeness A assume that they can’t have meant to do it, and therefore it must be fine to show up in tandem with their uninvited spouse or partner, thereby committing Rudeness B.
A lot of which could be easily avoided if people just asked themselves whichever of the following questions is relevant:
“If I don’t even know whether this person is partnered or not, am I really close enough to them to want to invite them to my wedding?”
“If this person sending the invitation doesn’t even know, or can’t be bothered to check, whether I’m partnered or not—or perhaps is just deliberately excluding my partner from the event due to personal antipathy or indifference—then am I really close enough to them to want to attend their wedding?”
It’s generally on the host to specify IMO; having said that, I did get caught out some years back when holding a BBQ on my allotment (like a community garden, but everyone gets a dedicated bit of it), and one of the guests showed up with their new partner (fine), their partner’s kids (also fine) and the dog (banned by site rules unless on a lead). I was not aware of the existence of the dog, so it hadn’t occurred to me to say.
Cultural expectations vary hugely, from ‘an invite to me includes me, partner and kids and any visitors’ all the way through to ‘person named only’.
We are dog people, and when we’ve invited people over for something casual, we’ll tell our dogged friends whether their dogs are invited or not. My brother usually showed up with his dogs no matter what, but they were always welcomed. For things that were not casual, we didn’t usually mention dogs in the invitation, and nobody has shown up with dog in tow to those events.
30+ years ago, one of my co-workers got married and invited everyone in the office, plus spouses for those who were married. I thought that was already pretty generous - she was not a wealthy person, and it wasn’t a gigantic wedding. One of our colleagues, however, had a longtime live-in partner, but because they were not married, her partner was not invited. She decided to RSVP that she was coming and then just…not show up. I thought that was pretty rude, and so did my co-workers, many of whom mentioned that they wished my long-term boyfriend at the time had been able to come instead, because he actually hung out with them socially on a regular basis and they liked him.
I can kind of see both sides of that one - the bride’s statement was that she had to draw the line somewhere, and if she didn’t think marriage meant something, she wouldn’t have been getting married. And it is super rude to RSVP that you are coming and then be a no-show.