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I had a disagreement with one of my groups recently: They believe that the identify spell is the only way to identify a magic item, while I believe a high enough Intelligence (Arcana) check is able to do the same.
What I came here to ask is: What are all the possible ways to identify a magic item?
In addition to using the Identify spell, you can identify a magic item during a short rest.
From the DMG:
[A]t the end of the short rest, the creature gains an intuitive understanding of how to activate any magical properties of the item, including any necessary command words.
The DMG describes the various ways to identify a Magic Item in the section "Identifying a Magic Item", on p. 136. We already have the TL;DR answer from A Very Large Bear, but I'm the "completionist" type. (And, to be fair, the question asks "All the possible ways to idenfity a magic item")
The identify spell is the fastest way to reveal an item's properties.
Alternatively, a character can focus on one magic item during a short rest, while being in physical contact with the item. At the end of the rest, the character learns the item's properties, as well as how to use them. Potions are an exception; a little taste is enough to tell the taster what the potion does.
Wearing or experimenting with an item can also offer hints about its properties. For example, if a character puts on a ring of jumping, you could say, "Your steps feel strangely springy." Perhaps the character then jumps up and down to see what happens. You then say the character jumps unexpectedly high.
Note that knowing that an item is a "ring of jumping" might not mean anything for the character, unless he has knowledge in magic items - simply knowing it makes him jump higher could be enough.
Sometimes a magic item carries a clue to its properties. The command word to activate a ring might be etched in tiny letters inside it, or a feathered design might suggest that it's a ring of feather falling.
Anything else your DM wants, or, if you are the DM, whatever you want. In particular, an Arcana check along experimentation or if the item is characteristic is a common (from my experience) house rule for acquiring full knowledge of the item. I have used Arcana checks to identify items; experimentation and identifying important features of the item before making the check would give advantage on the roll. Note that this usually makes the identify spell a lot less valuable, so, as a DM, make sure your Wizard (the character who probably has the spell) who spent 100 gp on a pearl to cast it is fine with house-ruling it this way.
Your group might be right if they are following the rule for harder identification of magic items, also described in the same section.
If you prefer magic items to have a greater mystique, consider removing the ability to identify the properties of a magic item during a short rest, and require the identify spell, experimentation, or both to reveal what a magic item does.
In addition to using an Identify spell, the following methods are also valid.
From DMG page 136:
...a character can focus on one magic item during a short rest, while being in physical contact with the item. At the end of the rest, the character learns the item's properties, as well as how to use them. Potions are an exception; a little taste is enough to tell the taster what the potion does.
There's a bit more about the short rest, but that's the gist of it.
Also from the DMG page 136:
Some magic items require a creature to form a bond with them before their magical properties can be used. This bond is called attunement, and certain items have a prerequisite for it. If the prerequisite is a class, a creature must be a member of that class to attune to the item. (If the class is a spellcasting class, a monster qualifies if that monster has spell slots and uses that class's spell list.)
Attunement is accomplish just like taking a short rest, but only the characters that meet the prerequisite can attune to the magic item.
Conclusion: Check the section in the DMG page 136 on Identifying a Magic Item and Attunement.
The PHB explicitly says you can identify magic items with an Arcana check (under a description of the intelligence attribute) - but provides no guidelines or DCs for doing so.
I'd recommend something like DC 10 arcana for a common magic item, DC 15 for an uncommon magic item, and DC 20+ for a rare or unique item, unless it's an artifact of repute, in which case knowing what it is should be easy, but knowing details about it might be more difficult.
This is a perennial problem with the 5e source material, in that it repeatedly contains examples of you being allowed to do things while providing no guidelines (PHB or DMG) for how a DM would set the check for doing so.
The DMG also has guidelines for experimenting with magic items to attempt to identify them. It also claims you can identify an object by spending a short rest focusing on it, or by trying to attune to it, or by casting identify on it. Obviously you could also have diaries or other documentation to help the players understand an item, or have an NPC tell them what it is.
We use the DCs from the DMG p238 for our Arcana skill checks to identify magic items. It's just slightly modified for magic item rarity and I let the Identify spell give Advantage on the roll. Plus it's home brewed to last for 1 minute in case they have several items to identify.
For reference, the skill check examples the DMG gives are as follows:
