Copenhagen

You can read the history of train ferries on the Copenhagen-Oslo route:
The end of the train ferries
All goods trains were removed from the route in November 1986, as two large goods train ferries began to operate between Helsingborg harhour's western port (Swedish: Västhamnen) and the North port (Danish: Nordhavn) in Copenhagen. The huge goods train ferries were sister ships, the Danish M/F Trekroner, and the Swedish M/S Öresund[53] were operated by DAN-link[54][55] until the opening of the Øresund Bridge in July 2000. But the passenger train line Copenhagen - Oslo continued to be using the DSB ferries across the HH route until the bridge opened 1 July 2000. Ironically, the so-called Linx trains between the Danish and Norwegian Capitals went into bankruptcy a year later.
https://en.wikipedia.org/wiki/HH_Ferry_route#The_end_of_the_train_ferries

AlexNL said:

Freight, and the SJ2000 service to Stockholm. There is a seasonal overnight service from Berlin, operated by Snälltåget.

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The night train to Berlin does not cross the bridge, and in addition there is also four daily X2000 to Gothenburg in the summer.

route101 said:

They must of done? Unless more recently , the Oresund ?

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Yes they did, until the Oresund bridge provided a non-ferry rail link. When I travelled between Denmark and Sweden years back, the normal route for international services was to run north from Copenhagen to Helsingør, then onto the ferry for the very short crossing to Helsingborg in Sweden, and on from there. (I'd forgotten, until seeing earlier stuff in this thread, that there were direct Denmark-Norway services via that train ferry too.) I remember the trains going on and off the boat being a very slick and speedy operation.

I'm surprised that there aren't now Copenhagen-Norway trains, nor even Malmo-Norway ones. Would there not be a market for the latter, at least, rather than the current slow connection with one or more changes?

Re "After that, there were some other cross-Sound train ferry routes but they weren't successful."
By "other" routes do you mean the H-H train ferry? If so, what does "not successful" mean? It surely ran for many years as the only Denmark-Sweden rail service; plenty of people used it when I was on it. I thought it only stopped when replaced by the Oresund bridge?

AlbertBeale said:

I'm surprised that there aren't now Copenhagen-Norway trains, nor even Malmo-Norway ones. Would there not be a market for the latter, at least, rather than the current slow connection with one or more changes?

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We might see Copenhagen-Oslo trains in the future. There two problems at the moment, lack of rolling stock. SJ has no spare trains, and we probably won't any new routes until new trains are delivered. And Gothenburg is a Terminus with very little free capacity, but there is currently a tunnel under construction that will change that.

JonasB said:

We might see Copenhagen-Oslo trains in the future. There two problems at the moment, lack of rolling stock. SJ has no spare trains, and we probably won't any new routes until new trains are delivered. And Gothenburg is a Terminus with very little free capacity, but there is currently a tunnel under construction that will change that.

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In addition to Gothenburg C capacity there are major structural problems on the Norwegian side. Oslo-Gothenburg is seen as worthwhile for high-speed trains but there are geographical limitations due to the narrow valleys.

While there is political support for high speed train in Norway the first investments are likely to be on domestic routes rather than a link to Sweden.
As part of its National Transportation Plan for 2018-2029, the Norwegian government has announced it will, on average, pump NOK 90 billion annually into investments, operations and maintenance of its roads and railways. Approximately NOK 120 billion is already slated to go into the rail industry for the period 2018-2023. This will cover new major projects, renewal and investments in existing infrastructure and digitalisation as Norway gears up to build railway systems to meet the needs of its growing population and reduce congestion and pollution on the roads.

In 1979 and again in 1988,I visited Gothenburg.The latter was an overnight and I was able to see the Hamburg to Oslo service at that time.
It ran in conjunction with a service to Stockholm.
There were 2 DB vehicles which ran through and incidentally were not advertised as calling any where in Denmark. The rest of the train was 5 Swedish and Norwegian coaches which did an out and back advertised as running only to Nassjo
There was also an overnight Oslo to Copenhagen service which did not call at Gothenburg Central but which changed locos in the yard outside. The loco from the Southbound Hamburg train returned to Oslo on the Northbound Copenhagen service and vice versa.
The Norwegian locos were Class E16.
In 1979 I saw ClassE14 on day trains.