Which is stronger: Threefish 1024-bit, SHACAL-2 512-bit, or AES-256? I found some App in the Google Store, and there were features only for the PRO version: Threefish 1024-bit and  SHACAL-2 512-bit. Meantime, AES-256 was available in the free version of app. 

So I concluded that the Threefish 1024-bit and SHACAL-2 512-bit algorithms are stronger, and I did a little research and discovered that AES-256 was invented in 2001, Threefish 1024 in 2008, and SHACAL-2 512 was also invented in 2001.

SHACAL-2 512 - I have read that this algorithm was something like "partly hacked for 50%" but for now considered as solid and unhackable.

Threefish 1024-bit - in the opinion of the authors who invented this algorithm it is more effective than AES. And it has a 2.9 security index, while AES has only a 1.7 security index (bigger is better).
Neither SHACAL-2 nor Threefish are "more" secure than AES because it's a case of "meh cannot break" with all of them.

All three ciphers are unbroken and all three use keys larger than 128-bit* meaning the bigger numbers in the key size have no value except to impress people with being bigger and with the company being able to waste computation time. Also note that in the most rares cases it is actually the crypto (and especially unlikely the block cipher) that is broken but rather the implementation and especially the data handling implementation.

Now for a bit of discussion on each cipher:


SHACAL-2. It is based on SHA-2's compression function and from what I can tell turning it into a block cipher was a case of "we did it because we could". It's practical deployment is absolutely negligible and cryptanalysis results are sparse, so I wouldn't trust it more than AES.
Threefish. Threefish was the block cipher underlying the Skein submission to the SHA-3 competition. It has some well-reputed names behind it and is a very useful tool block-cipher with a large block size that is especially hardened against related-key attacks and side-channel attacks. I would actually prefer Threefish over AES if you don't have a well-hardened implementation of AES available and you are on a 64-bit platform and you don't have hardware-accelerated AES available. This is because Threefish only uses additions, rotates and XORs (ARX) which makes it quite easy to implement securely and with solid performance. Analysis-wise it has seen quite a bit during the SHA-3 competition and a bit afterwards, but as Skein wasn't selected it hasn't seen much analysis since.
AES. AES is an excellent block cipher which has with-stood nearly two decades of intense cryptanalysis. Not only hasn't it been broken in this time, but we have also figured out how to implement it securely and it has wide hardware support these days, making it the performance- and security-wise best choice if available. If not, using a more modern ARX based cipher is also appropriate, but especially on a modern phone secure AES implementations should be available.


On this topic, also see the mandatory blog post by Matthew Green.



* If you believe that large-scale quantum computers will exist at some point, you need 256-bit symmetric keys to force $2^{128}$ quantum operations (thanks to Grover's Algorithm) which all three cipher support.