Usually, if you are displaying the Bookmarks Toolbar, the button for the drop-down list is on the right end of the bar, and the drop-down appears under it.

If you are not displaying the Bookmarks Toolbar, you can turn it on using

Alt+v (opens the classic View menu) > Toolbars > Bookmarks Toolbar

If the button is not in the expected location, has it moved to another location?

Please note that there are two similar looking buttons, and the other one opens and closes the Bookmarks sidebar on the left side of the web page. If you have the wrong button, use the customize feature to get the other one (drag from a toolbar to the dialog to remove a button, and drag from the dialog to a toolbar to add a button).

Alt+v (opens the classic View menu) > Toolbars > Customize

Any luck?

Note that the Bookmarks Menu button has a drop down marker if the customize window is open.
You can move the button to the left hand side of the Bookmarks Toolbar or to another toolbar (Navigation Toolbar or Tab bar).

Note that the presence of bookmarks (folder) with long names can cause the menu to open to the left.

I have read the above solution which makes no sense to me. I 'also didn't have a bookmark drop-down on the right hand side beside the home icon.

I have 2 pcs - 1 is a laptop which is set up properly so I checked the settings on it compared to the desk pc (which just had its hard drive changed so I lost everything) which didn't have the the drop down.

In order to get the drop down to the right, click on Firefox in the upper left corner & select options. In the section for toolbars ensure that only the navigation toolbar is selected. It worked for me. I don't know if it makes any difference or not but I also have selected, under options, tabs on top.

Hope this helps.

You only get the Bookmarks Menu button (with drop marker) on the Navigation Toolbar if the menu bar is hidden.
If the Bookmarks Toolbar is visible then the Bookmarks Menu button will be placed on that toolbar (possibly without drop marker on Windows).