JSOM

var listTitle = 'Tasks'
var itemId = 1;
var fileName = 'SharePoint User Guide.docx';


var ctx = SP.ClientContext.get_current();
var list = ctx.get_web().get_lists().getByTitle(listTitle);
var item = list.getItemById(itemId);
var attachmentFile = item.get_attachmentFiles().getByFileName(fileName); 
attachmentFile.deleteObject();
ctx.executeQueryAsync(
  function(){
     console.log('Attachment file has been deleted');  
  },
  function(sender,args) 
  {
     console.log(args.get_message());
  });

REST

function deleteAttachmentFile(webUrl,listTitle,itemId,fileName)
{
   return $.ajax({
      url: webUrl + "/_api/lists/getByTitle('" +  listTitle + "')/getItemById(" + itemId + ")/AttachmentFiles/getByFileName('" + fileName + "')",
      method: 'POST',
      contentType: 'application/json;odata=verbose',
      headers: {
        'X-RequestDigest': $('#__REQUESTDIGEST').val(),
        'X-HTTP-Method' : 'DELETE',
        'Accept': 'application/json;odata=verbose'
      }
   });    
}

Usage

deleteAttachmentFile(_spPageContextInfo.webAbsoluteUrl,'Tasks',1,'Instru ction.docx')
.done(function(data)
{
     console.log('Attachment file has been deleted');
})
.fail(
function(error){
    console.log(error.responseText);
});

JSOM

var listTitle = 'Tasks'
var itemId = 1;
var fileName = 'Instruction.docx';


var ctx = SP.ClientContext.get_current();
var list = ctx.get_web().get_lists().getByTitle(listTitle);
var item = list.getItemById(itemId);
var attachmentFile = item.get_attachmentFiles().getByFileName(fileName); 
attachmentFile.deleteObject();
ctx.executeQueryAsync(
  function(){
     console.log('Attachment file has been deleted');  
  },
  function(sender,args) 
  {
     console.log(args.get_message());
  });

REST

function deleteAttachmentFile(webUrl,listTitle,itemId,fileName)
{
   return $.ajax({
      url: webUrl + "/_api/lists/getByTitle('" +  listTitle + "')/getItemById(" + itemId + ")/AttachmentFiles/getByFileName('" + fileName + "')",
      method: 'POST',
      contentType: 'application/json;odata=verbose',
      headers: {
        'X-RequestDigest': $('#__REQUESTDIGEST').val(),
        'X-HTTP-Method' : 'DELETE',
        'Accept': 'application/json;odata=verbose'
      }
   });    
}

Usage

deleteAttachmentFile(_spPageContextInfo.webAbsoluteUrl,'Tasks',1,'Instruction.docx')
.done(function(data)
{
     console.log('Attachment file has been deleted');
})
.fail(
function(error){
    console.log(error.responseText);
});