JavaScript (Firefox 30+), 112 109 103 95 bytes

Look ma, no built-ins!

y=>[for(m of(i=0,y%4|y%400*!(y%100)&&6)+"63153042641")if((i++,y+(y>>2)-(y/100|0)*3/4|0)%7==m)i]

Here's a 107-byte ES6 version:

y=>[...(y%4|y%400*!(y%100)&&6)+"63153042641"].map((m,i)=>(y+(y>>2)-(y/100|0)*3/4|0)%7-m?0:i+1).filter(x=>x)

And here's my previous attempt, 123 113 bytes of ES6:

y=>[(l=y%4|y%400*!(y%100))?[7]:[1,7],[4,12],[9],[3,6],[8,11],[5],l?[1,2,10]:[2,10]][(y+(y>>2)-(y/100|0)*3/4|0)%7]

JavaScript (Firefox 30+), 68 bytes

y=>[for(m of Array(12).keys())if(new Date(y+400,++m).getDay()==2)m]

Batch, 160 152 bytes

@set/ay=%1,m=0,j=6*!(!(y%%4)*(y%%100)+(y%%400)),y=(y*5/4-y/100*3/4)%%7
@for %%d in (%j% 6 3 1 5 3 0 4 2 6 4 1)do @set/am+=1&if %%d==%y% call echo %%m%%

Port of @ETHproduction's answer. With month abbreviations for 197 189 bytes:

@set/ay=%1,j=6*!(!(y%%4)*(y%%100)+(y%%400)),y=(y*5/4-y/100*3/4)%%7
@for %%m in (Jan.%j% Feb.6 Mar.3 Apr.1 May.5 Jun.3 Jul.0 Aug.4 Sep.2 Oct.6 Nov.4 Dec.1)do @if %%~xm==.%y% call echo %%~nm

Perl+cal, 47 bytes

say`cal $_ $ARGV[0]`=~/\n.{5}\n/?$_:""for 1..12

Example:

$ perl -E 'say`cal $_ $ARGV[0]`=~/\n.{5}\n/?$_:""for 1..12' 2016

2







10


$

Python 2, 100 bytes

Ugh. Math with dates isn't as simple as I'd like.

lambda y:[m+1for m in range(12)if(date(y,12,31)if m>10else(date(y,m+2,1)-timedelta(1))).weekday()<1]

Try it online

Same length:

lambda y:[m-1for m in range(2,14)if(date(y,12,31)if m>12else(date(y,m,1)-timedelta(1))).weekday()<1]

J, 48 bytes

((2|2%~' '#@-.~_2,@{.4 5 7 8{"1])&>##\)@calendar

Uses the calendar builtin to generate an array of strings representing the months, then parses each string to determine the whether the last Monday is the last day of the month. It outputs each month as the month number of each. That is, Jan = 1, Feb = 2, ..., Dec = 12.

The output of calendar is

   _3 ]\ calendar 2016
┌─────────────────────┬─────────────────────┬─────────────────────┐
│         Jan         │         Feb         │         Mar         │
│ Su Mo Tu We Th Fr Sa│ Su Mo Tu We Th Fr Sa│ Su Mo Tu We Th Fr Sa│
│                 1  2│     1  2  3  4  5  6│        1  2  3  4  5│
│  3  4  5  6  7  8  9│  7  8  9 10 11 12 13│  6  7  8  9 10 11 12│
│ 10 11 12 13 14 15 16│ 14 15 16 17 18 19 20│ 13 14 15 16 17 18 19│
│ 17 18 19 20 21 22 23│ 21 22 23 24 25 26 27│ 20 21 22 23 24 25 26│
│ 24 25 26 27 28 29 30│ 28 29               │ 27 28 29 30 31      │
│ 31                  │                     │                     │
├─────────────────────┼─────────────────────┼─────────────────────┤
│         Apr         │         May         │         Jun         │
│ Su Mo Tu We Th Fr Sa│ Su Mo Tu We Th Fr Sa│ Su Mo Tu We Th Fr Sa│
│                 1  2│  1  2  3  4  5  6  7│           1  2  3  4│
│  3  4  5  6  7  8  9│  8  9 10 11 12 13 14│  5  6  7  8  9 10 11│
│ 10 11 12 13 14 15 16│ 15 16 17 18 19 20 21│ 12 13 14 15 16 17 18│
│ 17 18 19 20 21 22 23│ 22 23 24 25 26 27 28│ 19 20 21 22 23 24 25│
│ 24 25 26 27 28 29 30│ 29 30 31            │ 26 27 28 29 30      │
│                     │                     │                     │
├─────────────────────┼─────────────────────┼─────────────────────┤
│         Jul         │         Aug         │         Sep         │
│ Su Mo Tu We Th Fr Sa│ Su Mo Tu We Th Fr Sa│ Su Mo Tu We Th Fr Sa│
│                 1  2│     1  2  3  4  5  6│              1  2  3│
│  3  4  5  6  7  8  9│  7  8  9 10 11 12 13│  4  5  6  7  8  9 10│
│ 10 11 12 13 14 15 16│ 14 15 16 17 18 19 20│ 11 12 13 14 15 16 17│
│ 17 18 19 20 21 22 23│ 21 22 23 24 25 26 27│ 18 19 20 21 22 23 24│
│ 24 25 26 27 28 29 30│ 28 29 30 31         │ 25 26 27 28 29 30   │
│ 31                  │                     │                     │
├─────────────────────┼─────────────────────┼─────────────────────┤
│         Oct         │         Nov         │         Dec         │
│ Su Mo Tu We Th Fr Sa│ Su Mo Tu We Th Fr Sa│ Su Mo Tu We Th Fr Sa│
│                    1│        1  2  3  4  5│              1  2  3│
│  2  3  4  5  6  7  8│  6  7  8  9 10 11 12│  4  5  6  7  8  9 10│
│  9 10 11 12 13 14 15│ 13 14 15 16 17 18 19│ 11 12 13 14 15 16 17│
│ 16 17 18 19 20 21 22│ 20 21 22 23 24 25 26│ 18 19 20 21 22 23 24│
│ 23 24 25 26 27 28 29│ 27 28 29 30         │ 25 26 27 28 29 30 31│
│ 30 31               │                     │                     │
└─────────────────────┴─────────────────────┴─────────────────────┘

Usage

   f =: ((2|2%~' '#@-.~_2,@{.4 5 7 8{"1])&>##\)@calendar
   f 1
4 12
   f 297
5
   f 1776
9
   f 2000
1 7
   f 2016
2 10
   f 3385
1 2 10

Mathematica, 62 bytes

Position[a=#;DayName@DayRange[{#},{#+1},"EndOfMonth"],Monday]&

Anonymous function. Takes a number as input and returns a list of single-element lists of numbers as output. I'm honestly not sure myself how it works anymore.

Bash+cal, 58 bytes

$ cat t.sh
for A in {1..12};do cal $A $1|grep -qx .....&&echo $A;done
$ bash t.sh 2016
2
10
$

Ruby, 54 + 6 = 60 bytes

λ cat monday.rb
p (1..12).select{|m|Date.new($*[0].to_i,m,-1).monday?}
λ ruby -rdate monday.rb 2016
[2, 10]

6 bytes for -rdate on the command line to get the Date class from the standard library.

PHP, 159 bytes

while($z++<9999){$o=[];$m=0;while($m++<12)if(date("N",strtotime("$z-$m-".cal_days_in_month(0,$m,$z)))<2)$o[]=$m;echo count($o)>0?"$z:".implode(",",$o)."
":"";}

If running on Windows or a 32bit system there will be the dreaded 2038 bug, but on a 64bit linux system it's fine.

I did attempt to use date("t"... which is meant to represent the last date of the given month, but the results didn't match those previously mentioned in this thread.

Perl, 64 bytes

Includes +1 for -n

Give input on STDIN:

perl -M5.010 mon.pl <<< 2016

mon.pl:

#!/usr/bin/perl -n
map$b.=$/.gmtime$_.e4,-7e6..3e7;say$b=~/on (\S+ )\S.* $_.* 1 /g

MATL, 21 bytes

12:"G@QhO6(YO9XO77=?@

Months are displayed as numbers.

Try it online! Or verify all test cases.

Explanation

This uses date conversion builtin functions. For the given year it tests which months' last day is Monday.

Instead of explicitly specifying the last day of month k (which may be 28, 29, 30 or 31), we specify the 0-th day of month k+1, which is equivalent and does not depend on month or year.

12:      % Push [1 2 ... 12] (months)
"        % For each month k
  G      %   Push input
  @Q     %   Push k+1
  h      %   Concatenate
  O6(    %   Postpend four zeros. For example, for input 2016 and month k=1 
         %   (first iteration) this gives [2016 2 0 0 0 0] (year, month, day,
         %   hour, min, sec). The 0-th day of month k+1 is the same as the
         %   last day of month k.
  YO     %   Convert the above 6-element date vector to date number
  9XO    %   Convert date number to date string with output format 9, which 
         %   is weekday as a capital letter
  77=    %   Is it an 'M'?
  ?      %   If so
    @    %     Push current month (will be implicitly displayed)

Python 2, 94 bytes

from datetime import*
lambda y:[m for m in range(1,13)if date(y+(m>11),m%12+1,1).weekday()==1]

repl.it

An unnamed function, takes an integer year, outputs a list of the month numbers [1-12].

I also tried to beat the byte count with arithmetic without success (110 bytes). :

lambda y:map(lambda x,v:(23*((x+2)%13or 1)/9+y-2*(0<x<11)+(x>10)+v/4-v/100+v/400)%7==4,range(12),[y-1]+[y]*11)

An unnamed function which returns a list of boolean values representing if the months [Jan-Dec] end in a Monday

Bash + GNU utilities, 56 bytes

seq -f1month-1day$1-%g-1 12|date -f- +%B%u|sed -n s/1//p

Appears to require date version 8.25. The 8.23 version in Ideone doesn't cut it.

MySQL, 183 bytes

can probably be done shorter

SET @y=2016;CREATE TEMPORARY TABLE y (m INT);INSERT INTO y VALUES(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12);SELECT m FROM y WHERE 2=DAYOFWEEK(LAST_DAY(concat(@y,"-",m,"-01")))

Select a database. Replace 2016 with the desired year. Run.