The spell Magnificent Mansion states the following (emphasis mine) :
You conjure an extradimensional dwelling in range that lasts for the duration. You choose where its one entrance is located. The entrance shimmers faintly and is 5 feet wide and 10 feet tall. You and any creature you designate when you cast the spell can enter the extradimensional dwelling as long as the portal remains open. You can open or close the portal if you are within 30 feet of it. While closed, the portal is invisible.
The issue here is whether or not the presence of the portal can be detected while invisible, and if the nature of the spell can then be identified (or at the very least, identify that it manifests as a portal).
I first thought of Detect Magic, but the spell actually states (emphasis mine) :
For the duration, you sense the presence of magic within 30 feet of you. If you sense magic in this way, you can use your action to see a faint aura around any visible creature or object in the area that bears magic, and you learn its school of magic, if any.
As far as I understand it, this means that using Detect Magic, you would be able to sense there is some kind of magic around the portal's area, but you wouldn't be able to identify it further because it isn't visible (is a portal even an "object" in the first place, anyway?).
Is there no other way to detect the portal's presence while it is closed, or even do better than Detect Magic and identify the nature of the spell?
For a little bit of context, a player is thinking about very convoluted ways of creating an on-command trap, and using a Magnificent Mansion came up. Because of that, they've been asking whether or not it could be detected before it is tripped, and I haven't been able to find a solid answer to the question myself. I might cover the player's full plan in more depth in another question, if the need arises.
Best Answer
Anything that can see invisible things can discover it, if you consider it an object
The first step to get a real clue what is going on is that you need to be able to see the portal. There are multiple ways to make the portal visible: any creature with see invisibility, true seeing, truesight, or the ability to see invisible things (such as a Divination wizard with Third Eye set to that, or monsters that can see invisible things) would be able to see the portal, if it is an object.
The technical issue is that most of these abilities have the same wording, they allow you to see "invisible creatures or objects", and so they would not allow you to see the portal unless you consider it an object.
(As an aside: the way I run this in my home game is that if you can see invisible things, you can see anything invisible, not just creatures and objects. So at my table, it would be possible to see the invisible portal, but not to interact with it. This approach so far has not caused us any problems, and in fact my players would probably have some choice comments about it if I would argue they cannot see an invisible portal with see invisibility, due to a technicality of how the spell is worded.)
Is the portal an object?
I think it is ambiguous if the portal would count as an object. It is not further described, other than by its size and by saying it is invisible. Magical effects in general are not objects, but a door would be an object (it is even listed in the definition). I think it would be likely up to the DM, with my take being on the side of "not an object", because treating it as a normal object brings a lot of other problems and questions, such as if you can then attack it or break it open or not. If it is not an object, then you also cannot see it with abilities that only allow seeing objects, or determine its aura.
Detect magic
You can always sense the presence of magic with detect magic, you just cannot use your action to determine the nature of its aura if it is not an object you can see. So the portal can be detected, although not directly its exact location. Depending on available space, you can triangulate the location by moving around, and noting when you start to detect magic.
Detect magic does not allow you to identify the spell directly, even if you could see the portal and read the aura, it only gives you the school of magic. So, if you could see it, you would know it is a portal 5 feet wide and 10 tall, and if you could apply the second effect, that it is created by conjuration magic.
Finding out what spell caused the portal
If you can see the portal, the DM then might allow you to try and identify the spell creating such a portal, for example using an Intelligence (Arcana) check, but that would be a separate step.
For the purpose of knowing what you are dealing with, the information that a 5 feet by 10 feet invisible portal is a conjuration effect may be pretty immaterial -- how many spells are there that create such a portal? I think you could probably make a -- maybe slightly harder -- check to figure out what it is even without this information.