What causes “SYN to LISTEN sockets dropped”?

These counters ultimately come from the kernel and map to the LINUX_MIB_LISTENOVERFLOWS and LINUX_MIB_LISTENDROPS counters. You can see from the source of net/ipv4/tcp_ipv4.c(tcp_v4_syn_recv_sock) around line #1392 that when LINUX_MIB_LISTENOVERFLOWS is incremented, LINUX_MIB_LISTENDROPS will also be incremented but there are exit conditions where only the latter can be incremented so it's not a bug that they don't match.

In the same file you can see there's this code:

1291 int tcp_v4_conn_request(struct sock *sk, struct sk_buff *skb)
1292 {
1293         /* Never answer to SYNs send to broadcast or multicast */
1294         if (skb_rtable(skb)->rt_flags & (RTCF_BROADCAST | RTCF_MULTICAST))
1295                 goto drop;
1296 
1297         return tcp_conn_request(&tcp_request_sock_ops,
1298                                 &tcp_request_sock_ipv4_ops, sk, skb);
1299 
1300 drop:
1301         NET_INC_STATS_BH(sock_net(sk), LINUX_MIB_LISTENDROPS);
1302         return 0;
1303 }

So you can see at least one cause is a SYN to a broadcast or multicast address.

Usually wmem and rmem defaults are 212992 bytes. Apparently not enough on busy server. Raised to 8MB and the problem disappeared.

sysctl -w net.core.wmem_default=8388608
sysctl -w net.core.rmem_default=8388608