There are my core dumps!

As noted before, with systemd, your core dumps disappear into one of systemd's log sinks, instead of being stored to the current directory. By now, they are recoverable by normal users again.

TL;DR: Run coredumpctl gdb to invoke gdb with the most recent core dump.

Related bug about disappearing coredumps:

Scenario: A simple program that crashes with a segfault:

~/dumpcore$ cat dumpcore.c
#include <unistd.h>
void dump_core()   { int* boom = NULL; *boom = 2000; }
void its_time_to() { dump_core(); }
int main()         { its_time_to(); }
~/dumpcore$ cc -g -o dumpcore dumpcore.c
~/dumpcore$ ./dumpcore
Segmentation fault (core dumped)

Core dumps get written to systemd's core dumps directory (see man coredumpctl), and can be retrieved with the coredumpctl tool:

~/dumpcore$ coredumpctl list
TIME                            PID   UID   GID SIG PRESENT EXE
Sun 2016-10-02 14:44:14 CEST   3857  1000   100  11 * /home/me/dumpcore/dumpcore

By default, coredumpctl seems to use the most recent core dump, so that you can easily inspect the most recent core dump with:

~/dumpcore$ coredumpctl gdb
           PID: 3857 (dumpcore)
           UID: 1000 (me)
           GID: 100 (users)
        Signal: 11 (SEGV)
     Timestamp: Sun 2016-10-02 14:44:14 CEST (1min 3s ago)
  Command Line: ./dumpcore
    Executable: /home/me/dumpcore/dumpcore
 Control Group: /user.slice/user-1000.slice/user@1000.service/gnome-terminal-server.service
          Unit: user@1000.service
     User Unit: user@1000.service
         Slice: user-1000.slice
     Owner UID: 1000 (me)
       Boot ID: 73902fbce4e241f5bab007aad96895eb
    Machine ID: 3ff4d30a94914397acd6af26eed15114
      Hostname: limbo
      Coredump: /var/lib/systemd/coredump/core.dumpcore.1000.73902fbce4e241f5bab007aad96895eb.3857.1475412254000000000000.lz4
       Message: Process 3857 (dumpcore) of user 1000 dumped core.

                Stack trace of thread 3857:
                #0  0x00000000004004b6 dump_core (dumpcore)
                #1  0x00000000004004cd its_time_to (dumpcore)
                #2  0x00000000004004de main (dumpcore)
                #3  0x00007f89ba351291 __libc_start_main (
                #4  0x00000000004003da _start (dumpcore)

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Reading symbols from /home/me/dumpcore/dumpcore...done.
[New LWP 3857]

warning: Could not load shared library symbols for
Do you need "set solib-search-path" or "set sysroot"?
Core was generated by `./dumpcore'.
Program terminated with signal SIGSEGV, Segmentation fault.
#0  0x00000000004004b6 in dump_core () at dumpcore.c:2
2	void dump_core()   { int* boom = NULL; *boom = 2000; }
(gdb) bt
#0  0x00000000004004b6 in dump_core () at dumpcore.c:2
#1  0x00000000004004cd in its_time_to () at dumpcore.c:3
#2  0x00000000004004de in main () at dumpcore.c:4