How can I debug random Emacs server crashes?

I generally have a long-running instance of emacs --daemon, which I connect to with various emacsclients. Recently I have started experiencing random crashes. I say "random" because I haven't been able to figure out what causes the crashes. Often it happens e.g. when I leave my work computer over the weekend. However, I also remember at least one time when I started Emacs again after a crash, and it crashed again after a couple hours, during the same session.

I started an instance with emacs -Q (and then did M-x server-start) on my unused laptop, and it has not crashed yet. (Currently it has been: about a week.) One day I had need to use Emacs on that laptop, so I started up a separate session by doing emacs from a terminal window, and did not start the server. I left that session running, and when I came back a day or two later, I found that it had crashed, giving Segmentation fault. Thus, it would seem that this is not related to running Emacs in daemon mode.

I would do a binary search of my init file, but due to the randomness of the crashes, it would be a pain to go without much of the functionality I'm used to for an extended amount of time. Also, I wouldn't have a good way to know when I had fixed it, and the crashes weren't occuring.

I use Emacs 24.4, on Debian GNU/Linux 8, and on Ubuntu. (I have used Emacs on Windows before, but didn't use the --daemon, and didn't experience any crashes.)

How else can I debug these random crashes?

Edit: I get the same result when I launch Emacs and then do M-x server-start, and also when I use the Lucid toolkit.

I ran gdb emacs, and then at the (gdb) prompt did run, and this was the output:

Starting program: /usr/bin/emacs [Thread debugging using libthread_db_enabled] Using host libthread_db library "/lib/x86_64-linux-gnu/libthread_db.so.1". Warning: Cannot convert string "-*-courier-medium-r-*-*-*-120-*-*-*-*-io8859-*" to type FontStruct 

(Note that the last line also appears as terminal output when I launch Emacs outside of GDB.)

In Emacs, I did M-x server-start, and started work. It didn't crash all day, so I left it running over the weekend. When I got back today, I found Emacs unresponsive, and this additional output in GDB:

Program received signal SIGSEGV, Segmentation fault. mark_object (arg=42366066) at alloc.c:5601 5601 alloc.c: No such file or directory. 

After that line, control returned to the GDB prompt.

Edit 2: After building Emacs from source, I again ran it from GDB and did M-x server-start. Today I again found it unresponsive, and this was the output in GDB:

Starting program: [snip]/emacs-24.5/src/emacs [Thread debugging using libthread_db enabled] Using host libthread_db library "/lib/x86_64-linux-gnu/libthread_db.so.1". [New Thread 0x7fffee188700 (LWP 20124)] [New Thread 0x7fffe7df2700 (LWP 20125)] [New Thread 0x7fffe75f1700 (LWP 20126)] Program received signal SIGSEGV, Segmentation fault. mark_object (arg=47086818) at alloc.c:5973 5973 {