I had forgotten about the select by attributes link from the tables. Would be far easier if they were all located in the one place.
I did a search in the ESRI help documentation but that brought up Select by attributes as opposed to Select by attributes, which is what I required. Note that they are both called Select by attributes but they are not the same, although they look and feel very similar.
Perhaps ESRI should rename the table one Select by table attributes and the other Select by feature attributes, although may be that would still confuse users. I'm not going to look a ArcGIS Idea because I feel something like this won't be of interest to people/ESRI.
The option you want is actually two icons to the left when viewing the attribute table in George's screenshot. The very first icon start on the left has a drop-down arrow associated with it; click that and pick Select by Attributes. There is no difference "feature" attributes and "table" attributes, truthfully. While feature classes and tables do appear differently in the ArcMap table of contents, the Select by Attributes capability is exactly the same when being performed either on a feature class or on a table. Attributes can exist in tables or feature classes. The difference in HOW you access the Select by Attribute dialog box when dealing with a feature class versus a table is something to note though; for feature classes, you can get to the Select by Attribute dialog box by opening the Selection menu at the top of ArcMap. For tables, you have to open the attribute table first and use the button / drop-down arrow I've just mentioned. Both dialog boxes work the same way; but as you have correctly indicated, the Select by Attribute dialog box that gets opened from the ArcMap Selection menu does not work with tables.