Locked objects

As Kurt Gold ​said, check it up and remove the check.

Just to rule out a ghost object (on top and invisible within the screen view), which would be revealed by the expanded Layer in the Layers palette as suggested by Monika, and which would show in the Transform palette, can you select the object(s) by just Clicking?

If no other solution appears/works, you may try the list below, hoping for one of the easy ones at the top; you have not ruled out 4) and 6) by merely uninstalling and reinstalling.

If things that should work simply refuse to (all possibilities exhausted including the A) - C) below (and carefully making sure you are performing (all) the needed things)), you may try (the relevant part(s) of) the list 1) - 6) below.

Sometimes, (certain) things may fail or stop working for no apparent reason. When the (other) possible reasons/cures fail to work, it may be some kind of (temporay or permanent) corruption, or even some inconvenient preference setting(s), which may be cured with something on the following list set up in an attempt to provide a catchall solution for otherwise unsolvable cases. It starts with a few easy and harmless suggestions 1) and 2) for milder cases, and goes on with two alternative ways 3) and 4) of resetting preferences to the defaults (easily but irreversibly and more laboriously but more thoroughly and also reversibly), then follows a list 5) of various other possibilities, and it ends with a full reinstallation 6). If no other suggestions work, or if no other suggestions appear, you may start on the list and decide how far to go and/or which may be relevant.

The following is a general list of things you may try when

A) The issue is not in a specific file,

B) You have a printer correctly installed, connected, and turned on if it is physical printer (you may use Adobe PDF/Acrobat Distiller as the default printer with no need to have a printer turned on, obviously you will need to specify when you actually need to print on paper), and

C) It is not caused by issues with opening a file from external media.

D) It is not caused by other applications (you may close down/disable everything else running, including browsers and antivirus, then try again). There are quite a few usual suspects that may disturb and confuse Illy (job description Adobe Illustrator) so she moves in mysterious ways (even more than usual, some would say).

You may have tried/done some of them already; 1) and 2) are the easy ones for temporary strangenesses, and 3) and 4) are specifically aimed at possibly corrupt/inconvenient preferences); 5) is a list in itself, and 6) is the last resort.

If possible/applicable, you should save current artwork first, of course.

1) Close down Illy and open again;

2) Restart the computer (you may do that up to at least 5 times);

3) Close down Illy and press Ctrl+Alt+Shift/Cmd+Option+Shift during startup (easy but irreversible);

4) Move the folder (follow the link with that name) with Illy closed (more tedious but also more thorough and reversible), for CS3 - CC you may find the folder here:

https://helpx.adobe.com/illustrator/kb/preference-file-location-illustrator.html

5) Look through and try out the relevant among the Other options (follow the link with that name, Item 7) is a list of usual suspects among other applications that may disturb and confuse Illy, Item 15) applies to CC, CS6, and maybe CS5);

Even more seriously (this may be serious because you may need to restore plugins and whatnot afterwards if you have customized things), you may:

6) A) Uninstall (ticking the box to delete the preferences if applicable), B) run the Cleaner Tool (if you have CS3/CS4/CS5/CS6/CC), and C) reinstall. You may try without step B), but sometimes it is needed, because otherwise things may linger.

As it appears from this thread, it may be worth repeating this if it fails to help the first time:

https://forums.adobe.com/thread/2283530

To uninstall:

https://www.adobe.com/cfusion/search/index.cfm?loc=en_us&term=uninstall&cat=support&product=illustrator&self=1

Cleaner Tool:

http://www.adobe.com/support/contact/cscleanertool.html

Which version, which system?

Please show a screenshot with layers panel. Panel needs to be open and object in it visible.