If / when Schnorr signatures are deployed in a future witness version, it
may be possible to have non-interactive partial aggregation of the
signatures on a per-block basis.  This could save quite a bit of space.  It
*seems* not to have any security problems but this mailing list is very
good at finding vulnerabilities so that type of feedback is the main reason
I'm writing :) (A quick explanation of why this is horribly broken could
save me lots of time!)
(also sorry if this has been discussed; didn't see anything)

Quick recap / context of Schnorr sigs:

There are a bunch of private keys x1, x2, x3...
multiply by generator G to get x1G = P1, x2G = P2, x3G = P3

Everyone makes their sighash m1, m2, m3, and their random nonces k1, k2, k3.

To sign, people calculate s values:

s1 = k1 - h(m1, R1, P1)x1
s2 = k2 - h(m2, R2, P2)x2

(adding the P2 into the e hash value is not in most literature /
explanations but helps with some attacks; I beleive that's the current
thinking.  Anyway it doesn't matter for this idea)

Signature 1 is [R1, s1].  Verifiers check, given P1, m1, R1, s1:

s1G =? R1 - h(m1, R1, P1)P1

You can *interactively* make aggregate signatures, which requires
co-signers to build an aggregate R value by coming up with their own k
values, sharing their R with the co-signers, adding up the R's to get a
summed R, and using that to sign.

Non-interactively though, it seems like you can aggregate half the
signature.  The R values are unique to the [m, P] pair, but the s's can be
summed up:

s1 + s2 = k1 + k2 - h(m1, R1, P1)x1 - h(m2, R2, P2)x2

(s1 + s2)G = R1 + R2 - h(m1, R1, P1)P1 - h(m2, R2, P2)P2

To use this property in Bitcoin, when making transactions, wallets can sign
in the normal way, and the signature, consisting of [R, s] goes into the
witness stack.  When miners generate a block, they remove the s-value from
all compatible inputs, and commit to the aggregate s-value in the coinbase
transaction (either in a new OP_RETURN or alongside the existing witness
commitment structure).

The obvious advatage is that signatures go down to 32 bytes each, so you
can fit more of them in a block, and they take up less disk and network
space.  (In IBD; if a node maintains a mempool they'll need to receive all
the separate s-values)

Another advatage is that block verification is sped up.  For individual
signatures, the computation involves:

e = h(m1, R1, P1)           <- hash function, super fast
e*P                         <- point multiplication, slowest
R - e*P                     <- point addidion, pretty fast
s*G                         <- base point multiplication, pretty slow

with s-aggregate verification, the first three steps are still carried out
on each signature, but the s*G operation only needs to be done once.
Instead another point addition per signature is needed, where you have some
accumulator and add in the left side:
A += R - e*P
this can be parallelized pretty well as it's commutative.

The main downside I can see (assuming this actually works) is that it's
hard to cache signatures and quickly validate a block after it has come
in.  It might not be as bad as it first seems, as validation given chached
signatures looks possible without any elliptic curve operations.  Keep an
aggregate s-value (which is a scalar) for all the txs in your mempool.
When a block comes in, subtract all the s-values for txs not included in
the block.  If the block includes txs you weren't aware of, request them in
the same way compact blocks works, and get the full signature for those
txs.  It could be several thousand operations, but those are all bigInt
modular additions / subtractions which I believe are pretty quick in
comparison with point additions / multiplications.

There may be other complications due to the fact that the witness-txids
change when building a block.  TXIDs don't change though so should be
possible to keep track of things OK.

Also you can't "fail fast" for the signature verification; you have to add
everything up before you can tell if it's correct.  Probably not a big deal
as PoW check comes first, and invalid blocks are pretty uncommon and quite
costly.

Would be interested to hear if this idea looks promising.
Andrew Polestra mentioned something like this in the context of CT /
mimblewimble transactions a while ago, but it seems it may be applicable to
regular bitcoin Schnorr txs.

-Tadge