How do you delete a file you have selected in a Finder window, without using the mouse?
delete doesn't do it.
When I looked it up, I was said that command-delete is supposed to do it, but it doesn't work for me.
I had the same problem. Later I discovered that it was my own mistake. But also a mistake from Apple. On the extended keyboard with numeric pad there is a button says DELETE. (It's not on the short wireless keyboard) This is NOT the delete button for the combination Cmd-Delete. You have to use the button with the backarrow! So Cmd-Backarrow.
tl;dr: If ⌘-delete doesn't work, try ⌘-⌥-delete (command-option-delete) instead.
I had this problem on a new iMac with a standard Apple keyboard running the latest version of OS X (Mavericks 10.9.3 as of this writing). This was driving me crazy because I use ⌘-delete frequently, and I often switch between the iMac and my MacBook Air (running the same version of OS X) on which ⌘-delete works fine. I just discovered by chance and desperation that, for some reason, ⌘-⌥-delete (command-option-delete) does work. It's not as good as having the same keystrokes work on both machines, but at least now there's some keyboard shortcut I can use on the iMac.
UPDATE: For no apparent reason, ⌘-delete has started working.
This may not help many, but here is how I restored "Command+delete".
A while back, I was convinced that this key combination for "Move to Trash" was not working in my macOS Sierra. So, in System Preferences > Keyboard > Shortcuts > App Shortcuts, I attempted to manually define a shortcut "Move to Trash". But, I was never allowed to assign the same "Command+delete" key combination.
A frustrating few weeks have passed until this morning. But, this morning, I was determined to restore "Command+delete" and once again went into the App Shortcuts pane. There, it occurred to me to DELETE the very shortcut and key combination I had been trying to define. So, I selected my "Move to Trash" Finder shortcut and clicked on the "minus" button to delete it.
Upon returning to the Finder, I was stunned to see "Command+delete" appearing to the right of "Move to Trash" again.
So, if you have tried many things to restore the "Command+delete" key combination for "Move to Trash", one thing I would like for you to try is this: