Google N-gram Corpus Counts
╭─ーーーー─┬────────┬───────╮
│ 鼻たれ　 │ 16,730 │ 39.4% │ add
│ 鼻タレ　 │  8,546 │ 20.1% │ add
│ 洟垂れ　 │  6,598 │ 15.5% │
│ 鼻垂れ　 │  6,265 │ 14.8% │
│ はな垂れ │  4,036 │  9.5% │
│ 洟たれ　 │    218 │  0.5% │ add, sK 
│ 洟タレ　 │     38 │  0.1% │ add, sK
╰─ーーーー─┴────────┴───────╯

many visible forms, but i think they're useful

@@ -5 +5,4 @@
-<keb>はな垂れ</keb>
+<keb>鼻たれ</keb>
+</k_ele>
+<k_ele>
+<keb>鼻タレ</keb>
@@ -11,0 +15,11 @@
+</k_ele>
+<k_ele>
+<keb>はな垂れ</keb>
+</k_ele>
+<k_ele>
+<keb>洟たれ</keb>
+<ke_inf>&sK;</ke_inf>
+</k_ele>
+<k_ele>
+<keb>洟タレ</keb>
+<ke_inf>&sK;</ke_inf>

koj, daij, prog

meikyo has 鼻垂れ as well. sorted by hits

@@ -4,0 +4,3 @@
+<k_ele>
+<keb>はな垂れ</keb>
+</k_ele>
@@ -8,1 +11,1 @@
-<keb>はな垂れ</keb>
+<keb>鼻垂れ</keb>
@@ -15,3 +18,9 @@
-<gloss>tyke</gloss>
-<gloss>kid</gloss>
-<gloss>sniveling child</gloss>
+<gloss>runny-nosed child</gloss>
+<gloss>sniveler</gloss>
+<gloss>sniveller</gloss>
+</sense>
+<sense>
+<pos>&n;</pos>
+<misc>&derog;</misc>
+<gloss>greenhorn</gloss>
+<gloss>inexperienced person</gloss>

http://ejje.weblio.jp/content/洟垂れ  
http://dic.search.yahoo.co.jp/search?ei=UTF-8&p=はなたれ&fr=dic&stype=prefix

The JMDICT currently has 洟垂れ小僧, but it is also used without the 小僧.  I'm reading a book where the British-English "ickle Ronnie" becomes "洟垂れロニー", plus more examples at the reference links.