I don't know if there is a way to embed a control into the common grid so I do this by capturing a click event in a cell. You can put a picture/image into a cell if that makes it look better but just a word or two like "Select Something" will work.
Then show a ListGroup in BS or a list in jQuery, but my favorite is to turn another grid into a scrollable list. I position the list/grid where I want, usually centered near the top. This is not, of course, a classical drop list coming out of a cell in the grid but it gets the job done. If the pseudo drop list is long and needs scrolling it is a bit more complicated when using BS. It will need to be put into a container. This has been discussed in another topic elsewhere. I avoid doing this by using the grid list method.
One other trick I use is to simulate a modal list. This is done by showing and sizing a a label to the height and width of the form with the background color set to rgba(100, 100, 100, 0.4). An onclick event in this label can show a message like: "Select something from the list first." or just hide the list and label. This transparent label needs to be the next to last object in the Z order of the form and the grid or list control needs to be after it. One issue with a scroll-able grid over another scroll-able grid is that the scroll bar of the primary grid will "bleed through" and doesn't always look so good.