Q:

Looking for nonces of even numbers

Each one of bitcoin and its derived crypto-currencies has a nonce value in the block, no matter what the algorithm is. Every miner tries to search for a luck nonce which can make the hash value smaller than the target under the required difficulty.
However, recently I search scrypt based cryptocurrencies, like the dogecoin and vertcoin block chain for few blocks. I found most of the nonce are even numbers, except for (only manually browsed in block explorer)
Block #184161 - nonce = 8dce5c01
Block #184143 - nonce = 2a674001
Block #184139 - nonce = 930aa899
many other blocks from the latest block (Block #184174) and blocks in between of them are even numbers. Moreover, many nonce value are in hex number of form XXXXXX00 (in an integer hex number form, it is stored as 00XXXXXX in block), or multiples of 256.
The same outcome I observed in Vertcoin blocks. I manually traversed for few blocks and also found nonces of them are also even numbers.
I'd like to ask a question. If I configure the scrypt or n-scrypt to search only even numbers, is it possible to have higher chance to find the nonce which can solve the current quickly?
BTW, I won't expect the mining revenue of each miner in the pool (PPS or PPLNS) is going to be greater than that of normal nonce searching algorithm because the pool count for "shares" you found. When you skip odd numbers, you also lose the chance to get a share (solved by the odd nonce) which can meet the diff the pool gives to you. However, when a pool found a nonce that can solve the block, then the pool wins and gets the rewards.

Edited: Apr 18
I wrote a small program to collect some statistical data. From recent Dogecoin block #186,299 to #145,000 (the last mandatory update)
total 41,300 blocks

number of odds = 3,891  (9.42%)
number of evens = 37,409 (90.58%)

ratio of odd to even is about 1:10

Among the evens, the number of multiples of 256 = 35,106

85% of total
93.866% of evens

Update: 4/20
I recently also checked the nonces from block 552,780 to 253,898 of Litecoin. 
totally 298,883 blocks.

number of odds = 42,963 (14.374521%)
number of evens = 255,920 (85.625479%)
Among the evens, the number of multiples of 256 = 225,746

75.529890% of total

Update: 4/21
I use a small Perl script and call litecoind/dogecoind wallet to print out each block's nonce. It is very slow but quite simple. It would be very fast when you are using binary block database parser.
#!/usr/bin/perl -w

my $odd = 0;
my $even = 0;
my $m256 = 0;
my $total = 0;
for ($i = 186299; $i >= 145000; $i = $i-1) {
    $nonce = `dogecoind getblock \`dogecoind getblockhash $i\` | grep nonce`;
    chomp $nonce;
    $nonce =~ s/[^0-9]*//g;
    printf "%d %d\n", $i, $nonce;

    if (($nonce %2)== 1) {
            $odd++;
    }
    else {
            $even = $even + 1;
            $m256++ if (($nonce % 256) == 0);
    }
    $total ++;
}
printf "odds=%d (%f%%) evens=%d (%f%%) 256s=%d (%f%%)\n",
     $odd, (100.0*$odd/$total),
     $even, (100.0*$even/$total),
     $m256, 100.0*$m256/$total;

A:

I think that Tim S. may have the answer with his comment about endian-ness.  
Your observations about the nonce having its lowest byte zero (being a multiple of 256), are with respect to the little-endian byte order of the block itself.  From the perspective of a big-endian machine, these are statements about the high byte of the nonce.
So consider a miner which is big-endian.  The natural algorithm is "start with nonce=0, compute scrypt, increment nonce, repeat", so your "even" nonces will be tried first.  However, when a new transaction (or a new block from another miner) arrives, a new block header has to be constructed, and it would be natural to restart the nonce at zero when this happens.  In order to get a nonce that is "not a multiple of 256", it has to complete 2^24 hashes before being restarted.
Of course, x86 is the most common desktop CPU, and it is little-endian, but most scrypt mining is done on GPUs.  I hypothesize that a majority of these GPUs are big-endian, or at least that some common mining software causes them to increment their nonce in a big-endian manner.  Does anyone know if this is the case?
From this chart, it looks like modern GPUs can run scrypt at roughly 1 Mhash/sec.  So 2^24 hashes would take 16 seconds.  Litecoin is currently averaging roughly 10K transactions per day, which is an average of one every 8 seconds or so.  So it would not be surprising that a miner would  usually not get into their high byte (which for you is the low byte) before restarting.
This hypothesis would also explain why we do not see such a pattern with Bitcoin.  Current SHA-256 ASIC miners run at many Ghash/sec, and so are very likely to go through all 2^32 possible nonces before being restarted by new transaction data.  (We might see patterns in the extraNonce, though.) 

