This is a list of the possible Summons Screens that can be displayed when performing Summons. They do not affect the results in any way - quite the opposite, it's the result of the Summon that affects the animation - but some of these animations are able to reveal whether you pulled at least one SSR icon character or not.

Update: as of May 24th, 2018, all Multi-Summons performed with Dragon Stones now guarantee an SSR icon character in the 10th slot.

Update: as of June 25th, 2018, the Japanese version of the game has different animations as a result of the 3.13.0 update.

Animation Description
ByeGuys "Bye, Guys" is a rare summon animation which guarantees a character of SSR icon rarity.
OverFreeza "It's Over!!!! Freeza!!!" is an old animation screen which occasionally popped up guaranteeing a character of SR icon rarity. The animation was removed but was later brought back when the game was updated to version 3.10. The animation now guarantees SSR icon rarity pulls.
Summon Unkown "Thank You Dragon Balls!" is a rare summon screen which is similar to "Bye, Guys". It will guarantee an SSR icon character.
Pods Saiyan Pods are random animations during summoning. Anywhere from 0-7 pods may appear during summons. 7-pod summons guarantee an SSR icon rarity character (which is usually a Hercule Statue in Friend Summons, or an SR character Z-Awakened to SSR).
Hercule Hercule is another random animation that may appear if Saiyan Pods do not appear. Hercule may appear on the right or left side. Hercule does not have a direct effect on the rarity of a pull, but it is speculated that Hercule on the left side usually equals a higher rarity card.
PicsArt 01-31-12.02.33 Shenron is a very rare animation (the chance for it to appear is only 0.12%). It guarantees an SSR icon rarity character.
Rainbow Release

A rainbow release means that there's an increased chance to pull a featured character from the scouter display. If Goku stays in his base form after a rainbow release, it's guaranteed to shatter the screen for an SSR icon rarity card.

SS Animation If Goku transforms only to Super Saiyan, then this is generally only an SR icon rarity character, however, there is a rare chance to result in a shattered screen.
SS2 Animation If Goku transforms to Super Saiyan 2 during a summon, it guarantees a character of the SR icon rarity, although it can also be an SSR icon rarity character.
SS3 Animation If Goku transforms to Super Saiyan 3 during a summon, it guarantees an SSR icon rarity character.
Beeping Scouter A beeping scouter indicates that a character of the SSR icon rarity is present. This will not always occur with all SSR icon characters.

Animation Description
Oolong summon animation "Give Me Girl Panties!!!!!" is a summon animation that can appear after "Bye, Guys" or any of the other animations, by shattering the screen a second time. As far as we know, it guarantees an LR logo or a DF logo character.
Screenshot 20180625-004031 Before the Kamehameha blast, a number of different Z-Fighters may fly alongside Goku.

It looks like an LR logo or a DF logo character might be guaranteed whenever Majin Buu shows up while carrying Hercule on his back.

Screenshot 20180625-004333 On some occasions, it's possible that Goku will not ascend to Super Saiyan. Unless it's a multi-summon, the pull will likely be an R icon. There is, however, a rare chance for a shattered screen, producing an SSR icon instead.
20180625 175734 Occasionally, the text during a summon will have a rainbow color rather than the default yellow. Rainbow text means a higher possibility of a featured unit being in the summon. If Goku does not ascend and the text is rainbow, it is a guaranteed shattered screen.
Screenshot 20180625-004038 If Goku ascends only to Super Saiyan during a summon, at least one SR icon character is guaranteed. However, there is a rare chance to result in a shattered screen or an SSR icon.
Screenshot 20180625-004436 Goku may also ascend to Super Saiyan 2. This animation guarantees at least one SR icon, but like SS1, can give an SSR icon.
New SS3 animation If Goku ascends to Super Saiyan 3 during a summon, it guarantees an SSR icon character.
SSG Summon If Goku ascends past Super Saiyan 3, he will turn into Super Saiyan God. This animation guarantees 3 SSR icon characters.
SSB Goku animation Goku can ascend to Super Saiyan Blue Kaioken as a step above SSG, and most likely guarantees either an LR logo or a DF logo. However, this animation is currently unobtainable, and is only in the game files.
SV summon animation If Vegeta is flying alongside Goku before the Kamehameha blast, they'll have a chance to fuse into either Super Vegito or Vegito Blue. Super Vegito indicates that you're going to get either an LR logo or a DF logo character.
VB summon animation This animation guarantees an LR logo.
Summon card change The following silhouettes of R icon characters have a chance of being replaced by an LR logo or a DF logo character:


LR GokuBlack new summon screen When summoning a character that can be Dokkan Awakened to LR logo, the LR icon will appear over the card. Once the summon is over, the card will be highlighted GokuBlack summon highlight and the following text will appear:

Legendary character coming