There are many methods for manipulating living cells and pieces of dead ones. The method of choice is entirely context dependent.
Optical tweezers are highly concentrated laser beams which are used to attract and repel microscale and nanoscale objects in order to precisely move them. This method basically uses high powered photons to punch the cell around $\rightarrow$ manipulate its momentum.
Electrophoresis works on things that are charged. ABEL is a form of electrophoresis which requires active feedback. Electrophoresis implements the principle of electro-osmotic flow: flowing a mixed solution through porous material (gel electrophoresis) or capillaries (for capillary electrophoresis) to separate and purify the contents according to their electrophoretic mobility (a function of their molecular weight and charge). I won’t go into sequencing techniques in this post.
Note: For analyzing DNA or proteins with gel or capillary electrophoresis, you likely lysed your cell [killed it brutally] to extract the object of interest.
PAGE is a vertical electrophoretic technique used to uniformly move relatively small objects (lower molecular weight DNA, proteins., etc.). PAGE results in “smeared” bands compared to those resultant from agarose gel (due to the smaller pore size of polyacrylamide).
Cells have a non-uniform charge distribution. The solution holding the cells for observation often forms a counter-ion cloud which can shield and neutralize the charge of the cell. To avoid such complications, we use dielectrophoresis: an electrophoretic technique that works for all dielectric materials.
There are a variety of electrode geometries. As a general principle: the geometries are less complex for things on the order of cell size, (easy to trap), and more complex for smaller objects.
Other options include flowing cells through confocal microscopes, or magnetically manipulating cells using magnelles. Anything that generates a force reliably can be used as a trapping device!