Comments/Ratings for a Single Item

Sounds trivial, but I mention it:

*) You cannot drop a piece onto an occupied square (all pieces in all drop
variants I am aware of)

Restrictions on Check/Checkmate:

*) You cannot drop a piece giving checkmate (Shogi P)

Even more trivial, but:

General prohibition of dropping in cartain board positions, even in these,
to/from which it could normally move;

Attacking any opposing piece, not only the King.

Here are more ...

*) You cannot drop a piece to your 8th rank (all pieces in Pocket Mutation
Chess)

... and what about colourbound pieces? Any restrictions for keeping the
original colour binding or for not having two on the same colour? (I am not
aware of games stating such rules, but they look very natural to me)

Some CVs have drops to specific cells, such as Seirawan,
http://www.seirawanchess.com/,  Another example of that is great Altair, http://www.chessvariants.org/dpieces.dir/altair.html.  There piece goes to specific rank.  Lavieri accepted my calling the mechanism "drop" though in Altair the piece is not in a pocket, but already on board in a rank three away from the destination rank.  Jacks and Witches is another one of those, http://www.chessvariants.org/large.dir/contest84/jacksandwitches84.html,
where the off-board Jack is dropped as a move to one of four squares.

Set-up phases of placement CVs have their own rules for what are legitimate drops at the beginning: Vhttp://www.chessvariants.org/large.dir/bbchess/bbchess.html.

Very specialized: http://www.chessvariants.org/xiangqivariants.dir/para-xiangqi.html, because a dropped piece can go only orthogonal to guide piece (used before).  

( I think all these drops limited to area are better concept than long-time Shogi free drops with minor restrictions, at least for westerners.  Shogi is #1 played at GC, but same time why many players hate and won't play Shogi variants is that the pretty free drop ruins every well-thought position. )

Wasn't there any drop-game, where was restriction on having more than
certain number pieces? Or crtain number for each kind of pieces?

And also, players may have several different "places" of
"pieces-in-hand" - as in Hpstage chess there are own and opposing pieces,
and, of course, only friendly ones can be dropped.

Weak Square of the Jumping King,
http://www.chessvariants.org/38.dir/jumpingking.html, is drop Chess.  The
piece(s) in hand goes only to square the King is just vacating.

First of all, thanks for all your input! There were many things I would
never have thought of. Now some conclusions (mainly thinking aloud):

It would probably be good to treat 'gating', where introduction of a
piece takes places in combination with moving another piece, distinct from
true drops, which leave the remainder of the board unchanged. For gating
there is the choice to consider it as a side effect of the move of the
piece that was already on the board, and this seems the most convenient way
to do it. (In my Betza 2.0 proposal I had already introduced a move
modality 'u', which could unload something on the square indicated by the
Betza 'atom' to which it was prefixed.)

The true drops could be considered a special kind of Betza atom, and '@'
would be a good notation for it, as in standard algebraic notation drop
moves are written with it (e.g. P@c6). Dropping on empty squares only could
be written as m@, with the usual Betza modifier m. The power of Betza
notation is in clever choice of defaults, however. Where board moves almost
always have modality mc, which is thus default so it never has to be
written, it is very unusual to allow capture through drops. So for the @
atom, m could be defined as the default modality.

Then there is the concept of the 'drop zone'. In cases where this zone
consists of a few squares scattered over the board, there seems no
alternative to actually listing them. In Betza 2.0 I introduced the idea of
a 'limiter', which could be put on a modality modifier, like c{N,B}A for
a move that could only capture Knights and Bishops with an Alfil move. This
is a bit of an awkward notation, however, so for the more common cases,
where the drop zone is the entire board with the exception of some ranks at
the edges, I would like to have something simpler.

One method would be to write the range of ranks where the piece can be
dropped, e.g. 1-7 for Shogi Knights. This conflicts with the use of the
dash as chaining operator in Betza 2.0, however. It would be nice if we had
to write only a single rank number, for then we could write it behind the
atom: @7. This has similar syntax to W4 for a range-4 Rook. In fact it
would allow a similar interpretation, when we see the drop move as a move
that starts just in front of the board (on rank 0, as it were), on an
arbitrary file, and slides forward from there onto the board. The number
behind @ would then indicate the maximum rank it could reach. Like in the
Bex proposal for extension of Betza notation, a leading 0 on the number
could indicate it does not specify a maximum, but an exact number: @08
would mean 'drop on 8th rank only'.

Problem with this is that it is difficult to exempt a low rank from
dropping, as is needed for Crazyhouse Pawns, which can only be dropped on
ranks 2-7. This would then need an exhaustive listing @02@03@04@05@06@07,
which is pretty ugly. The problem is similar to that of 'skip-sliders'
like the Heavenly Tetrarchs of Tenjiku Shogi, which move as a slider after
skipping the first square (e.g. t(D,R) = D, then R in original Betza
notation). Another problem is that the dropped piece should be able to
reach any empty square within the range, even if there is blocking. IIRC
original Betza notation described a piece that could jump an arbitrary
number of obstacles with a 'jj' prefix. So mjj@7 would do it for Shogi
Knight drops.

Like the 'm', the 'jj' could be made default in absence of an explicit
single 'j' or 'n'. This offers the interesting possibility to specify
'non-jumping drops' n@8, which could drop anywhere upto the first
obstacle, but not beyond it. Such a drop mode was not mentioned here, but I
could imagine it would be a useful restriction on drops, to not make them
too aggressive. A drop on ranks 2-7 could be written with the Betza
'then' notation as t(@02,jjR5), i.e. initial drop on rank 2, followed by
an arbitrary jumping forward Rook move of upto 5 squares. Still not very
elegant.

There seems to be no obvious way to specify further restrictions, such as
on the number of Pawns per file, the number of color-bound pieces on the
same color, etc., in the Betza spirit. Perhaps some extra modifiers could
be introduced for such restrictions, but the available letters are already
almost exhausted. Because directional modifiers make no sense on a drop,
they could be reused for other purposes there. (It is already the case that
the meaning of directional modifiers depends on the atom that follows;
'fr' in frW means something different than in frF or frN.) So 'f' could
be taken to mean the drop is not allowed if there is already a piece of
that kind on the file, and 'b' that it is not allowed if it is already on
the board. 'ff' could be taken to mean there can be upto 2 on the file,
etc. 's' could similarly stand for 'same color square'.

This would make Shogi Pawn drops f@8, Shogi Knight drops @7, Tori-Shogi
Swallow drops ff@6. Hey, that is not so bad!

OK, after some thinking, I have decided upon the following notation for
future implementation in XBoard (to allow AI plugins to communicate the
rules to the GUI):

The possibility to drop a piece is indicated by putting the atom '@'
amongst its move. A numeric 'range' put on this atom indicates the depth
of the drop zone, starting at the 'home' rank. Default depth is 1 (i.e.
plain @ = @1, as usual in Betza notation), meaning dropping on the
home-rank only.

Modifiers f, s and b means that drops are only allowed when there isn't
another piece of the same type on the same file, same color (or hyper-color
for more stringent bindings) or board, respectively. If upto two pieces are
allowed per indicated area, the modifier is written twice, etc. This
actually covers all the setup rules for Big Board Chess: own board half
(@5), one Pawn per file (f@5), two Bishops per color (ss@5). Tori-Shogi
Swallow-Pawns would be fWff@6 (excluding the 7th rank where the Pawn would
have no moves), and a Shogi Knight would be fN@7 to exclude the last 2
ranks of the 9x9 board.

Default modality is 'm', for dropping on empty squares. If you can
capture with a drop, 'c' or 'mc' modifiers can be prefixed, as usual.

The modifier 'j' indicates drops on the home-rank are not allowed. So
Crazyhouse Pawns have a j@7 drop: from rank 2-7. This is consistent with
the interpretation that a drop is a move that slides the piece on the board
along a file from home edge, and the meaning XBoard will attach on a 'j'
prefix on B or R moves (for which the usual Betza meaning cannot apply, as
jW and jF are self-contradictory), which is skipping the first square of the slide.
Repetition of the 'j' could exclude more ranks.

Note that the sliding interpretation requires unlimited jumping of occupied
squares to be default for @. But this can be overruled by an explicit 'n'
modifier, which would forbid jumping anything, and thus force dropping
behind your least advanced piece in each file.

Drops that can be executed only in relation to other pieces, such as this
Castle in the Xiangqi variant, are better treated as moves of that Castle,
using Betza 2.0's 'unload' operator: uW-bW would move the Castle to a
neighboring square to unload something (from the hand, as it did not
capture anything before in that move), followed by stepping the Castle back
to its starting point (bW). Similarly Seirawan gatings can be described by
giving every other piece an extra move prefixed by "iuO-", e.g. N(iuO-N)
for the Knight. O means null move, so uO unloads a piece on the square of
origin, followed by a normal Knight move. And the 'i' indicates that only
virgin pieces have that move. No need to involve the drop atom here.