diff --git a/src/nvim/vterm/parser.c b/src/nvim/vterm/parser.c
index 79d348f2c1..f5f33c18c2 100644
--- a/src/nvim/vterm/parser.c
+++ b/src/nvim/vterm/parser.c
@@ -122,7 +122,7 @@ static void string_fragment(VTerm *vt, const char *str, size_t len, bool final)
   case CSI_INTERMED:
   case OSC_COMMAND:
   case DCS_COMMAND:
-    break;
+    return;
   }
 
   vt->parser.string_initial = false;
diff --git a/test/functional/terminal/parser_spec.lua b/test/functional/terminal/parser_spec.lua
index 67f47c7888..6c53957e6e 100644
--- a/test/functional/terminal/parser_spec.lua
+++ b/test/functional/terminal/parser_spec.lua
@@ -4,6 +4,10 @@ local clear = n.clear
 local api = n.api
 local assert_alive = n.assert_alive
 
+local OSC_PREFIX = string.char(0x1b, 0x5d)
+local BEL = string.char(0x07)
+local NUL = string.char(0x00)
+
 describe(':terminal', function()
   before_each(clear)
 
@@ -12,4 +16,48 @@ describe(':terminal', function()
     api.nvim_chan_send(chan, '\027]8;;https://example.com\027\\Example\027]8;;\027\n')
     assert_alive()
   end)
+
+  it('handles OSC-2 title setting', function()
+    -- OSC-2 should set title.
+    local chan = api.nvim_open_term(0, {})
+    local input = OSC_PREFIX .. '2;This title set with OSC 2' .. BEL
+    api.nvim_chan_send(chan, input)
+    --- @type string
+    local term_title = api.nvim_buf_get_var(0, 'term_title')
+    assert.Equal(term_title, 'This title set with OSC 2')
+    assert_alive()
+  end)
+
+  it('handles OSC-0 title and icon setting', function()
+    -- OSC-0 should set title and icon name to the same string. We currently ignore the icon name,
+    -- but the title should still be reflected.
+    local chan = api.nvim_open_term(0, {})
+    local input = OSC_PREFIX .. '0;This title set with OSC 0' .. BEL
+    api.nvim_chan_send(chan, input)
+    --- @type string
+    local term_title = api.nvim_buf_get_var(0, 'term_title')
+    assert.Equal(term_title, 'This title set with OSC 0')
+    assert_alive()
+  end)
+
+  it('handles control character following OSC prefix #34028', function()
+    local chan = api.nvim_open_term(0, {})
+    -- In order to test for the crash found in #34028 we need a ctrl char following the OSC_PREFIX
+    -- which causes `string_fragment()` to be called while in OSC_COMMAND mode, this caused
+    -- initial_string to be flipped back to false. At the end we need two more non-BEL control
+    -- characters, one to write into the 1 byte buffer, then another to trigger the callback one
+    -- more time so that realloc notices that it's internal data has been overwritten.
+    local input = OSC_PREFIX .. NUL .. '0;aaaaaaaaaaaaaaaaaaaaaaaaaaa' .. NUL .. NUL
+    api.nvim_chan_send(chan, input)
+    assert_alive()
+
+    -- On some platforms such as MacOS we need a longer string to reproduce the crash from #34028.
+    input = OSC_PREFIX .. NUL .. '0;'
+    for _ = 1, 256 do
+      input = input .. 'a'
+    end
+    input = input .. NUL .. NUL
+    api.nvim_chan_send(chan, input)
+    assert_alive()
+  end)
 end)